id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-10200 | Open Rational-Number Balls form Neighborhood Basis in Real Number Line | Let $\R$ be the real number line with the usual (Euclidean) metric.
Let $a \in \R$ be a point in $\R$.
Let $\BB_a$ be defined as:
:$\BB_a := \set {\map {B_\epsilon} a: \epsilon \in \Q_{>0} }$
that is, the set of all open $\epsilon$-balls of $a$ for rational $\epsilon$.
Then the $\BB_a$ is a basis for the neighborhood s... | Let $N$ be a neighborhood of $a$ in $M$.
Then by definition:
:$\exists \epsilon' \in \R_{>0}: \map {B_{\epsilon'} } a \subseteq N$
where $\map {B_{\epsilon'} } a$ is the open $\epsilon'$-ball at $a$.
From Open Ball in Real Number Line is Open Interval:
:$\map {B_{\epsilon'} } a = \openint {a - \epsilon'} {a + \epsilon'... | Let $\R$ be the [[Definition:Real Number Line with Euclidean Metric|real number line with the usual (Euclidean) metric]].
Let $a \in \R$ be a point in $\R$.
Let $\BB_a$ be defined as:
:$\BB_a := \set {\map {B_\epsilon} a: \epsilon \in \Q_{>0} }$
that is, the [[Definition:Set|set]] of all [[Definition:Open Ball|open ... | Let $N$ be a [[Definition:Neighborhood (Metric Space)|neighborhood]] of $a$ in $M$.
Then by definition:
:$\exists \epsilon' \in \R_{>0}: \map {B_{\epsilon'} } a \subseteq N$
where $\map {B_{\epsilon'} } a$ is the [[Definition:Open Ball|open $\epsilon'$-ball at $a$]].
From [[Open Ball in Real Number Line is Open Inter... | Open Rational-Number Balls form Neighborhood Basis in Real Number Line | https://proofwiki.org/wiki/Open_Rational-Number_Balls_form_Neighborhood_Basis_in_Real_Number_Line | https://proofwiki.org/wiki/Open_Rational-Number_Balls_form_Neighborhood_Basis_in_Real_Number_Line | [
"Real Intervals",
"Real Number Line with Euclidean Metric"
] | [
"Definition:Euclidean Metric/Real Number Line",
"Definition:Set",
"Definition:Open Ball",
"Definition:Rational Number",
"Definition:Basis for Neighborhood System"
] | [
"Definition:Neighborhood (Metric Space)",
"Definition:Open Ball",
"Open Ball in Real Number Line is Open Interval",
"Between two Real Numbers exists Rational Number",
"Open Real Interval is Open Ball",
"Definition:Open Ball",
"Subset Relation is Transitive",
"Open Ball is Neighborhood of all Points In... |
proofwiki-10201 | Open Reciprocal-N Balls form Neighborhood Basis in Real Number Line | Let $\R$ denote the real number line with the usual (Euclidean) metric.
Let $a \in \R$ be a point in $\R$.
Let $\BB_a$ be defined as:
:$\BB_a := \set {\map {B_\epsilon} a: \epsilon \in \set {\dfrac 1 n: n \in \N} }$
that is, the set of all open $\epsilon$-balls of $a$ for $\epsilon$ which are reciprocals of integers.
T... | Let $N$ be a neighborhood of $a$ in $M$.
Then by definition:
:$\exists \epsilon' \in \R_{>0}: \map {B_{\epsilon'} } a \subseteq N$
where $\map {B_{\epsilon'} } a$ is the open $\epsilon'$-ball at $a$.
From Open Ball in Real Number Line is Open Interval:
:$\map {B_{\epsilon'} } a = \openint {a - \epsilon'} {a + \epsilon'... | Let $\R$ denote the [[Definition:Real Number Line with Euclidean Metric|real number line with the usual (Euclidean) metric]].
Let $a \in \R$ be a point in $\R$.
Let $\BB_a$ be defined as:
:$\BB_a := \set {\map {B_\epsilon} a: \epsilon \in \set {\dfrac 1 n: n \in \N} }$
that is, the [[Definition:Set|set]] of all [[De... | Let $N$ be a [[Definition:Neighborhood (Metric Space)|neighborhood]] of $a$ in $M$.
Then by definition:
:$\exists \epsilon' \in \R_{>0}: \map {B_{\epsilon'} } a \subseteq N$
where $\map {B_{\epsilon'} } a$ is the [[Definition:Open Ball|open $\epsilon'$-ball at $a$]].
From [[Open Ball in Real Number Line is Open Inter... | Open Reciprocal-N Balls form Neighborhood Basis in Real Number Line | https://proofwiki.org/wiki/Open_Reciprocal-N_Balls_form_Neighborhood_Basis_in_Real_Number_Line | https://proofwiki.org/wiki/Open_Reciprocal-N_Balls_form_Neighborhood_Basis_in_Real_Number_Line | [
"Real Intervals",
"Real Number Line with Euclidean Metric"
] | [
"Definition:Euclidean Metric/Real Number Line",
"Definition:Set",
"Definition:Open Ball",
"Definition:Reciprocal",
"Definition:Integer",
"Definition:Basis for Neighborhood System"
] | [
"Definition:Neighborhood (Metric Space)",
"Definition:Open Ball",
"Open Ball in Real Number Line is Open Interval",
"Between two Real Numbers exists Rational Number",
"Definition:Rational Number/Canonical Form",
"Open Real Interval is Open Ball",
"Definition:Open Ball",
"Subset Relation is Transitive"... |
proofwiki-10202 | Subset of Open Reciprocal-N Balls forms Neighborhood Basis in Real Number Line | Let $\R$ denote the real number line with the usual (Euclidean) metric.
Let $a \in \R$ be a point in $\R$.
Let $k \in \Z$ be some fixed integer.
Let $\BB_a$ be defined as:
:$\BB_a := \set {\map {B_\epsilon} a: \epsilon \in \set {\dfrac 1 n: n \in \N, n > k} }$
that is, the set of all open $\epsilon$-balls of $a$ for $\... | Let $N$ be a neighborhood of $a$ in $M$.
Then by definition:
:$\exists \epsilon' \in \R_{>0}: \map {B_{\epsilon'} } a \subseteq N$
where $\map {B_{\epsilon'} } a$ is the open $\epsilon'$-ball at $a$.
From Open Ball in Real Number Line is Open Interval:
:$\map {B_{\epsilon'} } a = \openint {a - \epsilon'} {a + \epsilon'... | Let $\R$ denote the [[Definition:Real Number Line with Euclidean Metric|real number line with the usual (Euclidean) metric]].
Let $a \in \R$ be a point in $\R$.
Let $k \in \Z$ be some fixed [[Definition:Integer|integer]].
Let $\BB_a$ be defined as:
:$\BB_a := \set {\map {B_\epsilon} a: \epsilon \in \set {\dfrac 1 n... | Let $N$ be a [[Definition:Neighborhood (Metric Space)|neighborhood]] of $a$ in $M$.
Then by definition:
:$\exists \epsilon' \in \R_{>0}: \map {B_{\epsilon'} } a \subseteq N$
where $\map {B_{\epsilon'} } a$ is the [[Definition:Open Ball|open $\epsilon'$-ball at $a$]].
From [[Open Ball in Real Number Line is Open Inter... | Subset of Open Reciprocal-N Balls forms Neighborhood Basis in Real Number Line | https://proofwiki.org/wiki/Subset_of_Open_Reciprocal-N_Balls_forms_Neighborhood_Basis_in_Real_Number_Line | https://proofwiki.org/wiki/Subset_of_Open_Reciprocal-N_Balls_forms_Neighborhood_Basis_in_Real_Number_Line | [
"Real Intervals",
"Real Number Line with Euclidean Metric"
] | [
"Definition:Euclidean Metric/Real Number Line",
"Definition:Integer",
"Definition:Set",
"Definition:Open Ball",
"Definition:Reciprocal",
"Definition:Integer",
"Definition:Basis for Neighborhood System"
] | [
"Definition:Neighborhood (Metric Space)",
"Definition:Open Ball",
"Open Ball in Real Number Line is Open Interval",
"Between two Real Numbers exists Rational Number",
"Definition:Rational Number/Canonical Form",
"Open Real Interval is Open Ball",
"Definition:Open Ball",
"Subset Relation is Transitive"... |
proofwiki-10203 | Neighborhood Basis in Real Number Line is Infinite | Let $\R$ be the real number line with the usual (Euclidean) metric.
Let $a \in \R$ be a point in $\R$.
Let $\BB_a$ be a basis for the neighborhood system of $a$.
Then $\BB_a$ is an infinite set. | {{AimForCont}} $\BB_a$ be finite.
Let the elements of $\BB_a$ be enumerated as $N_1, N_2, \ldots, N_n$.
For each $N_k \in \BB_a$, let $\map {B_{\epsilon_k} } a$ be the open $\epsilon_k$-ball of $a$ for some $\epsilon_k \in \R_{>0}$.
Let $\alpha = \min \set {\epsilon_k: k \in \set {1, 2, \ldots, n} }$.
Consider the open... | Let $\R$ be the [[Definition:Real Number Line with Euclidean Metric|real number line with the usual (Euclidean) metric]].
Let $a \in \R$ be a point in $\R$.
Let $\BB_a$ be a [[Definition:Basis for Neighborhood System|basis for the neighborhood system]] of $a$.
Then $\BB_a$ is an [[Definition:Infinite Set|infinite s... | {{AimForCont}} $\BB_a$ be [[Definition:Finite Set|finite]].
Let the [[Definition:Element|elements]] of $\BB_a$ be [[Definition:Finite Enumeration|enumerated]] as $N_1, N_2, \ldots, N_n$.
For each $N_k \in \BB_a$, let $\map {B_{\epsilon_k} } a$ be the [[Definition:Open Ball|open $\epsilon_k$-ball]] of $a$ for some $\e... | Neighborhood Basis in Real Number Line is Infinite | https://proofwiki.org/wiki/Neighborhood_Basis_in_Real_Number_Line_is_Infinite | https://proofwiki.org/wiki/Neighborhood_Basis_in_Real_Number_Line_is_Infinite | [
"Real Intervals",
"Real Number Line with Euclidean Metric"
] | [
"Definition:Euclidean Metric/Real Number Line",
"Definition:Basis for Neighborhood System",
"Definition:Infinite Set"
] | [
"Definition:Finite Set",
"Definition:Element",
"Definition:Enumeration/Finite",
"Definition:Open Ball",
"Definition:Real Interval/Open",
"Definition:Neighborhood (Metric Space)",
"Definition:Subset",
"Definition:Basis for Neighborhood System"
] |
proofwiki-10204 | Distinct Points in Metric Space have Disjoint Neighborhoods | Let $M = \struct {A, d}$ be a metric space.
Let $x, y \in M: x \ne y$.
Then there exist neighborhoods $N_x$ and $N_y$ of $x$ and $y$ respectively such that $N_x \cap N_y = \O$, that is, that are disjoint. | Let $x, y \in A: x \ne y$.
From Distinct Points in Metric Space have Disjoint Open Balls, there exist disjoint open $\epsilon$-balls $\map {B_\epsilon} x$ and $\map {B_\epsilon} y$ containing $x$ and $y$ respectively.
From Open Ball is Neighborhood of all Points Inside it follows that $\map {B_\epsilon} x$ and $\map {B... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $x, y \in M: x \ne y$.
Then there exist [[Definition:Neighborhood (Metric Space)|neighborhoods]] $N_x$ and $N_y$ of $x$ and $y$ respectively such that $N_x \cap N_y = \O$, that is, that are [[Definition:Disjoint Sets|disjoint]]. | Let $x, y \in A: x \ne y$.
From [[Distinct Points in Metric Space have Disjoint Open Balls]], there exist [[Definition:Disjoint Sets|disjoint]] [[Definition:Open Ball of Metric Space|open $\epsilon$-balls]] $\map {B_\epsilon} x$ and $\map {B_\epsilon} y$ containing $x$ and $y$ respectively.
From [[Open Ball is Neighb... | Distinct Points in Metric Space have Disjoint Neighborhoods | https://proofwiki.org/wiki/Distinct_Points_in_Metric_Space_have_Disjoint_Neighborhoods | https://proofwiki.org/wiki/Distinct_Points_in_Metric_Space_have_Disjoint_Neighborhoods | [
"Neighborhoods",
"Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Neighborhood (Metric Space)",
"Definition:Disjoint Sets"
] | [
"Distinct Points in Metric Space have Disjoint Open Balls",
"Definition:Disjoint Sets",
"Definition:Open Ball",
"Open Ball is Neighborhood of all Points Inside",
"Definition:Neighborhood (Metric Space)"
] |
proofwiki-10205 | Open Ball in Cartesian Product under Chebyshev Distance | Let $M_1 = \struct {A_1, d_1}, M_2 = \struct {A_2, d_2}, \ldots, M_n = \struct {A_n, d_n}$ be metric spaces.
Let $\ds \AA = \prod_{i \mathop = 1}^n A_i$ be the cartesian product of $A_1, A_2, \ldots, A_n$.
Let $d_\infty: \AA \times \AA \to \R$ be the Chebyshev distance on $\AA$:
:$\ds \map {d_\infty} {x, y} = \max_{i \... | Let $\epsilon \in \R_{>0}$.
Let $x = \tuple {x_1, x_2, \ldots, x_n} \in \AA$.
Then:
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \map {B_\epsilon} {a; d_\infty}
| c =
}}
{{eqn | ll= \leadsto
| l = \map {d_\infty} {x, a}
| o = <
| r = \epsilon
| c = {{Defof|Open Ball of Metric S... | Let $M_1 = \struct {A_1, d_1}, M_2 = \struct {A_2, d_2}, \ldots, M_n = \struct {A_n, d_n}$ be [[Definition:Metric Space|metric spaces]].
Let $\ds \AA = \prod_{i \mathop = 1}^n A_i$ be the [[Definition:Finite Cartesian Product|cartesian product]] of $A_1, A_2, \ldots, A_n$.
Let $d_\infty: \AA \times \AA \to \R$ be the... | Let $\epsilon \in \R_{>0}$.
Let $x = \tuple {x_1, x_2, \ldots, x_n} \in \AA$.
Then:
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \map {B_\epsilon} {a; d_\infty}
| c =
}}
{{eqn | ll= \leadsto
| l = \map {d_\infty} {x, a}
| o = <
| r = \epsilon
| c = {{Defof|Open Ball of Metric... | Open Ball in Cartesian Product under Chebyshev Distance | https://proofwiki.org/wiki/Open_Ball_in_Cartesian_Product_under_Chebyshev_Distance | https://proofwiki.org/wiki/Open_Ball_in_Cartesian_Product_under_Chebyshev_Distance | [
"Chebyshev Distance",
"Open Balls"
] | [
"Definition:Metric Space",
"Definition:Cartesian Product/Finite",
"Definition:Chebyshev Distance",
"Definition:Open Ball"
] | [
"Definition:Set Equality/Definition 2"
] |
proofwiki-10206 | Point in Metric Space has Countable Neighborhood Basis | Let $M = \struct {A, d}$ be a metric space.
Let $a \in A$.
Then there exists a basis for the neighborhood system of $a$ which is countable. | Consider the countable set:
:$\BB_a := \set {\map {B_\epsilon} a: \exists n \in \Z_{>0}: \epsilon = \dfrac 1 n}$
That is, let $\BB_a$ be the set of all open $\epsilon$-balls of $a$ such that $\epsilon$ is of the form $\dfrac 1 n$ for (strictly) positive integral $n$.
Let $N$ be a neighborhood of $a$.
Then by definition... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $a \in A$.
Then there exists a [[Definition:Basis for Neighborhood System|basis for the neighborhood system]] of $a$ which is [[Definition:Countable Set|countable]]. | Consider the [[Definition:Countable Set|countable set]]:
:$\BB_a := \set {\map {B_\epsilon} a: \exists n \in \Z_{>0}: \epsilon = \dfrac 1 n}$
That is, let $\BB_a$ be the [[Definition:Set|set]] of all [[Definition:Open Ball of Metric Space|open $\epsilon$-balls]] of $a$ such that $\epsilon$ is of the form $\dfrac 1 n$ ... | Point in Metric Space has Countable Neighborhood Basis | https://proofwiki.org/wiki/Point_in_Metric_Space_has_Countable_Neighborhood_Basis | https://proofwiki.org/wiki/Point_in_Metric_Space_has_Countable_Neighborhood_Basis | [
"Neighborhoods"
] | [
"Definition:Metric Space",
"Definition:Basis for Neighborhood System",
"Definition:Countable Set"
] | [
"Definition:Countable Set",
"Definition:Set",
"Definition:Open Ball",
"Definition:Strictly Positive/Integer",
"Definition:Neighborhood (Metric Space)",
"Definition:Neighborhood (Metric Space)",
"Between two Real Numbers exists Rational Number",
"Definition:Rational Number/Canonical Form",
"Definitio... |
proofwiki-10207 | Neighborhood Basis in Cartesian Product under Chebyshev Distance | Let $M_1 = \struct {A_1, d_1}, M_2 = \struct {A_2, d_2}, \ldots, M_n = \struct {A_n, d_n}$ be metric spaces.
Let $\ds \AA = \prod_{i \mathop = 1}^n A_i$ be the cartesian product of $A_1, A_2, \ldots, A_n$.
Let $d_\infty: \AA \times \AA \to \R$ be the Chebyshev distance on $\AA$:
:$\ds \map {d_\infty} {x, y} = \max_{i \... | Let $N$ be a neighborhood of $a$ in $M$.
Then by definition of neighborhood:
:$\exists \epsilon \in \R_{>0}: \map {B_\epsilon} {a; d_\infty} \subseteq N$
where $\map {B_\epsilon} {a; d_\infty}$ is the open $\epsilon$-ball of $a$ under $d_\infty$.
From Open Ball in Cartesian Product under Chebyshev Distance:
:$\ds \map ... | Let $M_1 = \struct {A_1, d_1}, M_2 = \struct {A_2, d_2}, \ldots, M_n = \struct {A_n, d_n}$ be [[Definition:Metric Space|metric spaces]].
Let $\ds \AA = \prod_{i \mathop = 1}^n A_i$ be the [[Definition:Finite Cartesian Product|cartesian product]] of $A_1, A_2, \ldots, A_n$.
Let $d_\infty: \AA \times \AA \to \R$ be the... | Let $N$ be a [[Definition:Neighborhood (Metric Space)|neighborhood]] of $a$ in $M$.
Then by definition of [[Definition:Neighborhood (Metric Space)|neighborhood]]:
:$\exists \epsilon \in \R_{>0}: \map {B_\epsilon} {a; d_\infty} \subseteq N$
where $\map {B_\epsilon} {a; d_\infty}$ is the [[Definition:Open Ball of Metric... | Neighborhood Basis in Cartesian Product under Chebyshev Distance | https://proofwiki.org/wiki/Neighborhood_Basis_in_Cartesian_Product_under_Chebyshev_Distance | https://proofwiki.org/wiki/Neighborhood_Basis_in_Cartesian_Product_under_Chebyshev_Distance | [
"Chebyshev Distance",
"Neighborhoods"
] | [
"Definition:Metric Space",
"Definition:Cartesian Product/Finite",
"Definition:Chebyshev Distance",
"Definition:Basis for Neighborhood System",
"Definition:Cartesian Product/Finite",
"Definition:Basis for Neighborhood System"
] | [
"Definition:Neighborhood (Metric Space)",
"Definition:Neighborhood (Metric Space)",
"Definition:Open Ball",
"Open Ball in Cartesian Product under Chebyshev Distance",
"Open Ball is Neighborhood of all Points Inside",
"Definition:Neighborhood (Metric Space)",
"Definition:Basis for Neighborhood System",
... |
proofwiki-10208 | Projection from Cartesian Product under Chebyshev Distance is Continuous | Let $M_1 = \struct {A_1, d_1}, M_2 = \struct {A_2, d_2}, \ldots, M_n = \struct {A_n, d_n}$ be metric spaces.
Let $\ds \AA = \prod_{i \mathop = 1}^n A_i$ be the cartesian product of $A_1, A_2, \ldots, A_n$.
Let $d_\infty: \AA \times \AA \to \R$ be the Chebyshev distance on $\AA$:
:$\ds \map {d_\infty} {x, y} = \max_{i \... | Let $\epsilon \in \R_{>0}$.
Let $a = \tuple {a_1, a_2, \ldots, a_n} \in \AA$.
Let $\map {B_\epsilon} {a_i; d_i}$ be the open $\epsilon$-ball of $a_i$ in $M_i$.
From Open Ball in Cartesian Product under Chebyshev Distance:
:$\ds \map {B_\epsilon} {a; d_\infty} = \prod_{i \mathop = 1}^n \map {B_\epsilon} {a_i; d_i}$
By d... | Let $M_1 = \struct {A_1, d_1}, M_2 = \struct {A_2, d_2}, \ldots, M_n = \struct {A_n, d_n}$ be [[Definition:Metric Space|metric spaces]].
Let $\ds \AA = \prod_{i \mathop = 1}^n A_i$ be the [[Definition:Finite Cartesian Product|cartesian product]] of $A_1, A_2, \ldots, A_n$.
Let $d_\infty: \AA \times \AA \to \R$ be the... | Let $\epsilon \in \R_{>0}$.
Let $a = \tuple {a_1, a_2, \ldots, a_n} \in \AA$.
Let $\map {B_\epsilon} {a_i; d_i}$ be the [[Definition:Open Ball of Metric Space|open $\epsilon$-ball]] of $a_i$ in $M_i$.
From [[Open Ball in Cartesian Product under Chebyshev Distance]]:
:$\ds \map {B_\epsilon} {a; d_\infty} = \prod_{i \... | Projection from Cartesian Product under Chebyshev Distance is Continuous | https://proofwiki.org/wiki/Projection_from_Cartesian_Product_under_Chebyshev_Distance_is_Continuous | https://proofwiki.org/wiki/Projection_from_Cartesian_Product_under_Chebyshev_Distance_is_Continuous | [
"Chebyshev Distance",
"Projections",
"Continuous Mappings on Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Cartesian Product/Finite",
"Definition:Chebyshev Distance",
"Definition:Projection (Mapping Theory)",
"Definition:Continuous Mapping (Metric Space)/Space"
] | [
"Definition:Open Ball",
"Open Ball in Cartesian Product under Chebyshev Distance",
"Definition:Projection (Mapping Theory)",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Continuous Mapping (Metric Space)/Point",
"Definition:Continuous Mapping (Metric Space)/Point",
"Definition:Continuous ... |
proofwiki-10209 | Continuity of Mapping to Cartesian Product under Chebyshev Distance | Let $M_1 = \struct {A_1, d_1}, M_2 = \struct {A_2, d_2}, \ldots, M_n = \struct {A_n, d_n}$ be metric spaces.
Let $\ds \AA = \prod_{i \mathop = 1}^n A_i$ be the cartesian product of $A_1, A_2, \ldots, A_n$.
Let $d_\infty: \AA \times \AA \to \R$ be the Chebyshev distance on $\AA$:
:$\ds \map {d_\infty} {x, y} = \max_{i \... | {{WLOG}}, let $i \in \set {1, 2, \ldots, n}$ be arbitrary. | Let $M_1 = \struct {A_1, d_1}, M_2 = \struct {A_2, d_2}, \ldots, M_n = \struct {A_n, d_n}$ be [[Definition:Metric Space|metric spaces]].
Let $\ds \AA = \prod_{i \mathop = 1}^n A_i$ be the [[Definition:Finite Cartesian Product|cartesian product]] of $A_1, A_2, \ldots, A_n$.
Let $d_\infty: \AA \times \AA \to \R$ be the... | {{WLOG}}, let $i \in \set {1, 2, \ldots, n}$ be arbitrary. | Continuity of Mapping to Cartesian Product under Chebyshev Distance | https://proofwiki.org/wiki/Continuity_of_Mapping_to_Cartesian_Product_under_Chebyshev_Distance | https://proofwiki.org/wiki/Continuity_of_Mapping_to_Cartesian_Product_under_Chebyshev_Distance | [
"Chebyshev Distance",
"Projections",
"Continuous Mappings on Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Cartesian Product/Finite",
"Definition:Chebyshev Distance",
"Definition:Projection (Mapping Theory)",
"Definition:Metric Space",
"Definition:Mapping",
"Definition:Continuous Mapping (Metric Space)/Space",
"Definition:Continuous Mapping (Metric Space)/Space"
] | [] |
proofwiki-10210 | Positive Image of Point of Continuous Real Function implies Positive Closed Interval of Domain | Let $f: \R \to \R$ be a continuous real function.
Let $a \in \R$ such that $\map f a > 0$.
Then:
:$\exists k \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in \closedint {a - \delta} {a + \delta}: \map f x \ge k$ | Let $\map f a = l$ where $l > 0$.
As $f$ is continuous:
:$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \size {y - x} < \delta \implies \size {\map f y - \map f x} < \epsilon$
Let $\epsilon = \dfrac l 2 = k$.
Then:
:$\exists \delta' \in \R_{>0}: \forall y \in \R: \size {y - x} < \delta' \implies \size {\map... | Let $f: \R \to \R$ be a [[Definition:Continuous Real Function|continuous real function]].
Let $a \in \R$ such that $\map f a > 0$.
Then:
:$\exists k \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in \closedint {a - \delta} {a + \delta}: \map f x \ge k$ | Let $\map f a = l$ where $l > 0$.
As $f$ is [[Definition:Continuous Real Function|continuous]]:
:$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \size {y - x} < \delta \implies \size {\map f y - \map f x} < \epsilon$
Let $\epsilon = \dfrac l 2 = k$.
Then:
:$\exists \delta' \in \R_{>0}: \forall y \in \R: \... | Positive Image of Point of Continuous Real Function implies Positive Closed Interval of Domain | https://proofwiki.org/wiki/Positive_Image_of_Point_of_Continuous_Real_Function_implies_Positive_Closed_Interval_of_Domain | https://proofwiki.org/wiki/Positive_Image_of_Point_of_Continuous_Real_Function_implies_Positive_Closed_Interval_of_Domain | [
"Continuous Functions"
] | [
"Definition:Continuous Real Function"
] | [
"Definition:Continuous Real Function"
] |
proofwiki-10211 | Limit of Sequence in Metric Space in Neighborhood | Let $M = \struct {A, d}$ be a metric space.
Let $\sequence {a_n}$ be a sequence in $A$.
Then $\ds \lim_{n \mathop \to \infty} a_n = a$ {{iff}} for each neighborhood $V$ of $a$:
:$\exists N \in \N: n > N \implies a_n \in V$ | === Necessary Condition ===
Let $\ds \lim_{n \mathop \to \infty} a_n = a$.
Let $V$ be a neighborhood of $a$.
By definition of neighborhood:
:$\exists \epsilon \in \R_{>0}: \map {B_\epsilon} a \subseteq V$
where $\map {B_\epsilon} a$ denotes the open $\epsilon$-ball of $a$ in $M$.
By definition of limit:
:$\exists N \in... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $\sequence {a_n}$ be a [[Definition:Sequence|sequence in $A$]].
Then $\ds \lim_{n \mathop \to \infty} a_n = a$ {{iff}} for each [[Definition:Neighborhood (Metric Space)|neighborhood]] $V$ of $a$:
:$\exists N \in \N: n > N \implies a_n \in V$ | === Necessary Condition ===
Let $\ds \lim_{n \mathop \to \infty} a_n = a$.
Let $V$ be a [[Definition:Neighborhood (Metric Space)|neighborhood]] of $a$.
By definition of [[Definition:Neighborhood (Metric Space)|neighborhood]]:
:$\exists \epsilon \in \R_{>0}: \map {B_\epsilon} a \subseteq V$
where $\map {B_\epsilon} a... | Limit of Sequence in Metric Space in Neighborhood | https://proofwiki.org/wiki/Limit_of_Sequence_in_Metric_Space_in_Neighborhood | https://proofwiki.org/wiki/Limit_of_Sequence_in_Metric_Space_in_Neighborhood | [
"Metric Spaces",
"Limits of Sequences",
"Neighborhoods"
] | [
"Definition:Metric Space",
"Definition:Sequence",
"Definition:Neighborhood (Metric Space)"
] | [
"Definition:Neighborhood (Metric Space)",
"Definition:Neighborhood (Metric Space)",
"Definition:Open Ball",
"Definition:Limit of Sequence/Metric Space",
"Definition:Neighborhood (Metric Space)",
"Definition:Neighborhood (Metric Space)",
"Definition:Limit of Sequence/Metric Space"
] |
proofwiki-10212 | Square of Real Number is Non-Negative | Let $x \in \R$.
Then:
: $0 \le x^2$ | There are two cases to consider:
: $(1): \quad x = 0$
: $(2): \quad x \ne 0$
Let $x = 0$.
Then:
:$x^2 = 0$
and thus
:$0 \le x^2$
{{qed|lemma}}
Let $x \ne 0$.
From Square of Non-Zero Real Number is Strictly Positive it follows that:
:$0 < x^2$
and so by definition:
:$0 \le x^2$
{{Qed}} | Let $x \in \R$.
Then:
: $0 \le x^2$ | There are two cases to consider:
: $(1): \quad x = 0$
: $(2): \quad x \ne 0$
Let $x = 0$.
Then:
:$x^2 = 0$
and thus
:$0 \le x^2$
{{qed|lemma}}
Let $x \ne 0$.
From [[Square of Non-Zero Real Number is Strictly Positive]] it follows that:
:$0 < x^2$
and so by definition:
:$0 \le x^2$
{{Qed}} | Square of Real Number is Non-Negative | https://proofwiki.org/wiki/Square_of_Real_Number_is_Non-Negative | https://proofwiki.org/wiki/Square_of_Real_Number_is_Non-Negative | [
"Square Function",
"Real Analysis"
] | [] | [
"Square of Non-Zero Real Number is Strictly Positive"
] |
proofwiki-10213 | Real Number between Zero and One is Greater than Square | Let $x \in \R$.
Let $0 < x < 1$.
Then:
:$0 < x^2 < x$ | We are given that $0 < x < 1$.
By direct application of Real Number Ordering is Compatible with Multiplication, it follows that:
: $0 \times x < x \times x < 1 \times x$
and the result follows.
{{Qed}} | Let $x \in \R$.
Let $0 < x < 1$.
Then:
:$0 < x^2 < x$ | We are given that $0 < x < 1$.
By direct application of [[Real Number Ordering is Compatible with Multiplication]], it follows that:
: $0 \times x < x \times x < 1 \times x$
and the result follows.
{{Qed}} | Real Number between Zero and One is Greater than Square/Proof 1 | https://proofwiki.org/wiki/Real_Number_between_Zero_and_One_is_Greater_than_Square | https://proofwiki.org/wiki/Real_Number_between_Zero_and_One_is_Greater_than_Square/Proof_1 | [
"Real Number between Zero and One is Greater than Square",
"Square Function",
"Real Analysis",
"Inequalities"
] | [] | [
"Real Number Ordering is Compatible with Multiplication"
] |
proofwiki-10214 | Real Number between Zero and One is Greater than Square | Let $x \in \R$.
Let $0 < x < 1$.
Then:
:$0 < x^2 < x$ | We have that Real Numbers form Ordered Integral Domain.
Thus Square of Element Less than Unity in Ordered Integral Domain applies directly.
{{qed}} | Let $x \in \R$.
Let $0 < x < 1$.
Then:
:$0 < x^2 < x$ | We have that [[Real Numbers form Ordered Integral Domain]].
Thus [[Square of Element Less than Unity in Ordered Integral Domain]] applies directly.
{{qed}} | Real Number between Zero and One is Greater than Square/Proof 2 | https://proofwiki.org/wiki/Real_Number_between_Zero_and_One_is_Greater_than_Square | https://proofwiki.org/wiki/Real_Number_between_Zero_and_One_is_Greater_than_Square/Proof_2 | [
"Real Number between Zero and One is Greater than Square",
"Square Function",
"Real Analysis",
"Inequalities"
] | [] | [
"Real Numbers form Ordered Integral Domain",
"Square of Element Less than Unity in Ordered Integral Domain"
] |
proofwiki-10215 | Real Number Greater than One is Less than Square | Let $x \in \R$.
Let $x > 1$.
Then:
:$x^2 > x$ | As $x > 1$ it follows that $x > 0$.
Thus by Real Number Ordering is Compatible with Multiplication:
: $x \times x > 1 \times x$
and the result follows.
{{Qed}} | Let $x \in \R$.
Let $x > 1$.
Then:
:$x^2 > x$ | As $x > 1$ it follows that $x > 0$.
Thus by [[Real Number Ordering is Compatible with Multiplication]]:
: $x \times x > 1 \times x$
and the result follows.
{{Qed}} | Real Number Greater than One is Less than Square | https://proofwiki.org/wiki/Real_Number_Greater_than_One_is_Less_than_Square | https://proofwiki.org/wiki/Real_Number_Greater_than_One_is_Less_than_Square | [
"Square Function",
"Real Analysis"
] | [] | [
"Real Number Ordering is Compatible with Multiplication"
] |
proofwiki-10216 | Power of Real Number greater than One is Unbounded Above | Let $x \in \R$ be a real number such that $x > 1$.
Let set $S = \set {x^n: n \in \N}$.
Then $S$ is unbounded above. | {{AimForCont}} $S$ were bounded above.
Then $S$ has a supremum $B$.
As $x > 1$, it follows that $\dfrac B x < B$ and so therefore $\dfrac B x$ can not be an upper bound.
Therefore:
:$\exists n \in \N: x^n > \dfrac B x \implies x^{n + 1} > B$
So $B$ can not be an upper bound.
From that contradiction it can be concluded ... | Let $x \in \R$ be a [[Definition:Real Number|real number]] such that $x > 1$.
Let [[Definition:Set|set]] $S = \set {x^n: n \in \N}$.
Then $S$ is [[Definition:Unbounded Above Set|unbounded above]]. | {{AimForCont}} $S$ were [[Definition:Bounded Above Set|bounded above]].
Then $S$ has a [[Definition:Supremum of Set|supremum]] $B$.
As $x > 1$, it follows that $\dfrac B x < B$ and so therefore $\dfrac B x$ can not be an [[Definition:Upper Bound of Set|upper bound]].
Therefore:
:$\exists n \in \N: x^n > \dfrac B x \... | Power of Real Number greater than One is Unbounded Above | https://proofwiki.org/wiki/Power_of_Real_Number_greater_than_One_is_Unbounded_Above | https://proofwiki.org/wiki/Power_of_Real_Number_greater_than_One_is_Unbounded_Above | [
"Powers"
] | [
"Definition:Real Number",
"Definition:Set",
"Definition:Bounded Above Set/Unbounded"
] | [
"Definition:Bounded Above Set",
"Definition:Supremum of Set",
"Definition:Upper Bound of Set",
"Definition:Upper Bound of Set",
"Definition:Contradiction",
"Definition:Upper Bound of Set",
"Proof by Contradiction",
"Definition:Bounded Above Set"
] |
proofwiki-10217 | Power of Real Number between Zero and One is Bounded | Let $x \in \R$ be a real number.
Let $0 < x < 1$.
Let set $S = \set {x^n: n \in \N}$.
Then:
:$\inf S = 0$
and:
:$\sup S = 1$
where $\inf S$ and $\sup S$ are the infimum and supremum of $S$ respectively. | When $n = 0$ it follows that $x^n = 1$ and so $\sup S \ge 1$.
Let $x^k \in S$.
As $x < 1$, it follows from Real Number Ordering is Compatible with Multiplication that:
:$x^{k + 1} < x^k$
So:
:$\forall x \in S: x \le 1$
Hence it follows that $\sup S = 1$.
As $x > 0$, it follows by Real Number Ordering is Compatible with... | Let $x \in \R$ be a [[Definition:Real Number|real number]].
Let $0 < x < 1$.
Let [[Definition:Set|set]] $S = \set {x^n: n \in \N}$.
Then:
:$\inf S = 0$
and:
:$\sup S = 1$
where $\inf S$ and $\sup S$ are the [[Definition:Infimum of Set|infimum]] and [[Definition:Supremum of Set|supremum]] of $S$ respectively. | When $n = 0$ it follows that $x^n = 1$ and so $\sup S \ge 1$.
Let $x^k \in S$.
As $x < 1$, it follows from [[Real Number Ordering is Compatible with Multiplication]] that:
:$x^{k + 1} < x^k$
So:
:$\forall x \in S: x \le 1$
Hence it follows that $\sup S = 1$.
As $x > 0$, it follows by [[Real Number Ordering is Co... | Power of Real Number between Zero and One is Bounded | https://proofwiki.org/wiki/Power_of_Real_Number_between_Zero_and_One_is_Bounded | https://proofwiki.org/wiki/Power_of_Real_Number_between_Zero_and_One_is_Bounded | [
"Powers"
] | [
"Definition:Real Number",
"Definition:Set",
"Definition:Infimum of Set",
"Definition:Supremum of Set"
] | [
"Real Number Ordering is Compatible with Multiplication",
"Real Number Ordering is Compatible with Multiplication",
"Definition:Lower Bound of Set",
"Definition:Lower Bound of Set",
"Definition:Upper Bound of Set",
"Power of Real Number greater than One is Unbounded Above",
"Definition:Bounded Above Set... |
proofwiki-10218 | Limit of Function by Convergent Sequences/Real Number Line | Let $f$ be a real function defined on an open interval $\openint a b$, except possibly at the point $c \in \openint a b$.
Then $\ds \lim_{x \mathop \to c} \map f x = l$ {{iff}}:
::for each sequence $\sequence {x_n}$ of points of $\openint a b$ such that $\forall n \in \N_{>0}: x_n \ne c$ and $\ds \lim_{n \mathop \to \i... | === Necessary Condition ===
Let $\ds \lim_{x \mathop \to c} \map f x = l$.
Let $\epsilon \in \R_{>0}$.
Then by the definition of the limit of a real function:
:$\exists \delta \in \R_{>0}: \size {\map f x - l} < \epsilon$
provided $0 < \size {x - c} < \delta$.
Now suppose that $\sequence {x_n}$ is a sequence of element... | Let $f$ be a [[Definition:Real Function|real function]] defined on an [[Definition:Open Real Interval|open interval]] $\openint a b$, except possibly at the point $c \in \openint a b$.
Then $\ds \lim_{x \mathop \to c} \map f x = l$ {{iff}}:
::for each [[Definition:Real Sequence|sequence]] $\sequence {x_n}$ of points ... | === Necessary Condition ===
Let $\ds \lim_{x \mathop \to c} \map f x = l$.
Let $\epsilon \in \R_{>0}$.
Then by the definition of the [[Definition:Limit of Real Function|limit of a real function]]:
:$\exists \delta \in \R_{>0}: \size {\map f x - l} < \epsilon$
provided $0 < \size {x - c} < \delta$.
Now suppose that ... | Limit of Function by Convergent Sequences/Real Number Line | https://proofwiki.org/wiki/Limit_of_Function_by_Convergent_Sequences/Real_Number_Line | https://proofwiki.org/wiki/Limit_of_Function_by_Convergent_Sequences/Real_Number_Line | [
"Limits of Real Functions",
"Limits of Sequences"
] | [
"Definition:Real Function",
"Definition:Real Interval/Open",
"Definition:Real Sequence"
] | [
"Definition:Limit of Real Function",
"Definition:Real Sequence",
"Definition:Element",
"Definition:Limit of Real Function",
"Definition:Real Sequence",
"Definition:Element",
"Definition:Real Sequence",
"Definition:Element"
] |
proofwiki-10219 | Continuity of Mapping between Metric Spaces by Convergent Sequence | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.
Let $f: A_1 \to A_2$ be a mapping.
Then $f$ is continuous at $a \in X$ {{iff}}:
:whenever $\ds \lim_{n \mathop \to \infty} x_n = a$ for a sequence $\sequence {x_n}$ of points of $A_1$
it is true that:
:$\ds \lim_{n \mathop \to \infty} \ma... | === Necessary Condition ===
Let $f$ be continuous at $a \in A_1$.
Let $\ds \lim_{n \mathop \to \infty} x_n = a$.
Let $V$ be a neighborhood of $\map f a$.
Then by Metric Space Continuity by Inverse of Mapping between Neighborhoods $f^{-1} \sqbrk V$ is a neighborhood of $a$.
By Limit of Sequence in Metric Space in Neighb... | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be [[Definition:Metric Space|metric spaces]].
Let $f: A_1 \to A_2$ be a [[Definition:Mapping|mapping]].
Then $f$ is [[Definition:Continuous at Point of Metric Space|continuous]] at $a \in X$ {{iff}}:
:whenever $\ds \lim_{n \mathop \to \infty} x_n = a$ for... | === Necessary Condition ===
Let $f$ be [[Definition:Continuous at Point of Metric Space|continuous]] at $a \in A_1$.
Let $\ds \lim_{n \mathop \to \infty} x_n = a$.
Let $V$ be a [[Definition:Neighborhood (Metric Space)|neighborhood]] of $\map f a$.
Then by [[Metric Space Continuity by Inverse of Mapping between Neig... | Continuity of Mapping between Metric Spaces by Convergent Sequence | https://proofwiki.org/wiki/Continuity_of_Mapping_between_Metric_Spaces_by_Convergent_Sequence | https://proofwiki.org/wiki/Continuity_of_Mapping_between_Metric_Spaces_by_Convergent_Sequence | [
"Continuous Mappings on Metric Spaces",
"Limits of Sequences"
] | [
"Definition:Metric Space",
"Definition:Mapping",
"Definition:Continuous Mapping (Metric Space)/Point",
"Definition:Sequence"
] | [
"Definition:Continuous Mapping (Metric Space)/Point",
"Definition:Neighborhood (Metric Space)",
"Metric Space Continuity by Inverse of Mapping between Neighborhoods",
"Definition:Neighborhood (Metric Space)",
"Limit of Sequence in Metric Space in Neighborhood",
"Definition:Neighborhood (Metric Space)",
... |
proofwiki-10220 | Limit of Image of Sequence/Real Number Line | Let $f$ be a real function which is continuous on the interval $\Bbb I$.
Let $\sequence {x_n}$ be a sequence of points in $\Bbb I$ such that:
:$\ds \lim_{n \mathop \to \infty} x_n = \xi$
where:
:$(1): \quad \xi \in \Bbb I$
:$(2): \quad \ds \lim_{n \mathop \to \infty} x_n$ denotes the limit of $x_n$.
Then:
:$\ds \lim_{n... | From Limit of Real Function by Convergent Sequences, we have:
:$\ds \lim_{x \mathop \to \xi} \map f x = l$
{{iff}}:
:for each sequence $\sequence {x_n}$ of points of $\openint a b$ such that:
::$\forall n \in \N_{>0}: x_n \ne \xi$
:and:
::$\ds \lim_{n \mathop \to \infty} x_n = \xi$
:it is true that:
::$\ds \lim_{n \mat... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Continuous on Interval|continuous]] on the [[Definition:Real Interval|interval]] $\Bbb I$.
Let $\sequence {x_n}$ be a [[Definition:Real Sequence|sequence]] of points in $\Bbb I$ such that:
:$\ds \lim_{n \mathop \to \infty} x_n = \xi$
where:
... | From [[Limit of Real Function by Convergent Sequences]], we have:
:$\ds \lim_{x \mathop \to \xi} \map f x = l$
{{iff}}:
:for each [[Definition:Real Sequence|sequence]] $\sequence {x_n}$ of points of $\openint a b$ such that:
::$\forall n \in \N_{>0}: x_n \ne \xi$
:and:
::$\ds \lim_{n \mathop \to \infty} x_n = \xi$
:it... | Limit of Image of Sequence/Real Number Line | https://proofwiki.org/wiki/Limit_of_Image_of_Sequence/Real_Number_Line | https://proofwiki.org/wiki/Limit_of_Image_of_Sequence/Real_Number_Line | [
"Limits of Sequences",
"Real Analysis"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Interval",
"Definition:Real Interval",
"Definition:Real Sequence",
"Definition:Limit of Sequence/Real Numbers"
] | [
"Limit of Function by Convergent Sequences/Real Number Line",
"Definition:Real Sequence",
"Definition:Continuous Real Function/Interval"
] |
proofwiki-10221 | Infimum of Subset of Real Numbers is Arbitrarily Close | Let $A \subseteq \R$ be a subset of the real numbers.
Let $b$ be an infimum of $A$.
Let $\epsilon \in \R_{>0}$.
Then:
:$\exists x \in A: x - b < \epsilon$ | Note that $A$ is non-empty as the empty set does not admit an infimum (in $\R$).
Suppose $\epsilon \in \R_{>0}$ such that:
:$\forall x \in A: x - b \ge \epsilon$
Then:
:$\forall x \in A: b + \epsilon \le x$
and so $b + \epsilon$ would be a lower bound of $A$ which is greater than $b$.
But since $b$ is an infimum of $A$... | Let $A \subseteq \R$ be a [[Definition:Subset|subset]] of the [[Definition:Real Number|real numbers]].
Let $b$ be an [[Definition:Infimum of Subset of Real Numbers|infimum]] of $A$.
Let $\epsilon \in \R_{>0}$.
Then:
:$\exists x \in A: x - b < \epsilon$ | Note that $A$ is [[Definition:Non-Empty Set|non-empty]] as the [[Definition:Empty Set|empty set]] does not admit an [[Definition:Infimum of Subset of Real Numbers|infimum]] (in $\R$).
Suppose $\epsilon \in \R_{>0}$ such that:
:$\forall x \in A: x - b \ge \epsilon$
Then:
:$\forall x \in A: b + \epsilon \le x$
and so ... | Infimum of Subset of Real Numbers is Arbitrarily Close | https://proofwiki.org/wiki/Infimum_of_Subset_of_Real_Numbers_is_Arbitrarily_Close | https://proofwiki.org/wiki/Infimum_of_Subset_of_Real_Numbers_is_Arbitrarily_Close | [
"Real Analysis"
] | [
"Definition:Subset",
"Definition:Real Number",
"Definition:Infimum of Set/Real Numbers"
] | [
"Definition:Non-Empty Set",
"Definition:Empty Set",
"Definition:Infimum of Set/Real Numbers",
"Definition:Lower Bound of Set/Real Numbers",
"Definition:Infimum of Set/Real Numbers",
"Proof by Contradiction"
] |
proofwiki-10222 | Existence of Sequence in Set of Real Numbers whose Limit is Infimum | Let $A \subseteq \R$ be a non-empty subset of the real numbers.
Let $b$ be an infimum of $A$.
Then there exists a sequence $\sequence {a_n}$ in $\R$ such that:
:$(1): \quad \forall n \in \N: a_n \in A$
:$(2): \quad \ds \lim_{n \mathop \to \infty} a_n = b$ | From Infimum of Subset of Real Numbers is Arbitrarily Close:
For $\epsilon = \dfrac 1 n$ there exists an $a_n \in A$ such that:
:$a_n - b < \dfrac 1 n$
Since $b$ is an infimum of $A$:
:$0 \le a_n - b$
Therefore:
:$\ds \lim_{n \mathop \to \infty} a_n = b$
{{qed}} | Let $A \subseteq \R$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of the [[Definition:Real Number|real numbers]].
Let $b$ be an [[Definition:Infimum of Subset of Real Numbers|infimum]] of $A$.
Then there exists a [[Definition:Real Sequence|sequence]] $\sequence {a_n}$ in $\R$ such that:
:... | From [[Infimum of Subset of Real Numbers is Arbitrarily Close]]:
For $\epsilon = \dfrac 1 n$ there exists an $a_n \in A$ such that:
:$a_n - b < \dfrac 1 n$
Since $b$ is an [[Definition:Infimum of Subset of Real Numbers|infimum]] of $A$:
:$0 \le a_n - b$
Therefore:
:$\ds \lim_{n \mathop \to \infty} a_n = b$
{{qed}} | Existence of Sequence in Set of Real Numbers whose Limit is Infimum | https://proofwiki.org/wiki/Existence_of_Sequence_in_Set_of_Real_Numbers_whose_Limit_is_Infimum | https://proofwiki.org/wiki/Existence_of_Sequence_in_Set_of_Real_Numbers_whose_Limit_is_Infimum | [
"Real Analysis",
"Limits of Sequences"
] | [
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Real Number",
"Definition:Infimum of Set/Real Numbers",
"Definition:Real Sequence"
] | [
"Infimum of Subset of Real Numbers is Arbitrarily Close",
"Definition:Infimum of Set/Real Numbers"
] |
proofwiki-10223 | Existence of Sequence in Subset of Metric Space whose Limit is Infimum | Let $M = \struct {A, d}$ be a metric space.
Let $a \in A$.
Let $S \subseteq A$ be a non-empty subset of $A$.
Then there exists a sequence $\sequence {a_n}$ of points of $S$ such that:
:$\ds \lim_{n \mathop \to \infty} \map d {a, a_n} = \map d {a, S}$ | From Existence of Sequence in Set of Real Numbers whose Limit is Infimum:
:$\ds \lim_{n \mathop \to \infty} \map d {a, a_n} = b$
where $b$ is an infimum of $\map d {a, a_n}$.
Hence the result by definition of distance to a subset of a metric space.
{{qed}} | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $a \in A$.
Let $S \subseteq A$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of $A$.
Then there exists a [[Definition:Sequence|sequence]] $\sequence {a_n}$ of points of $S$ such that:
:$\ds \lim_{n \mathop \to \in... | From [[Existence of Sequence in Set of Real Numbers whose Limit is Infimum]]:
:$\ds \lim_{n \mathop \to \infty} \map d {a, a_n} = b$
where $b$ is an [[Definition:Infimum of Subset of Real Numbers|infimum]] of $\map d {a, a_n}$.
Hence the result by definition of [[Definition:Distance between Element and Subset of Metr... | Existence of Sequence in Subset of Metric Space whose Limit is Infimum | https://proofwiki.org/wiki/Existence_of_Sequence_in_Subset_of_Metric_Space_whose_Limit_is_Infimum | https://proofwiki.org/wiki/Existence_of_Sequence_in_Subset_of_Metric_Space_whose_Limit_is_Infimum | [
"Metric Spaces",
"Limits of Sequences"
] | [
"Definition:Metric Space",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Sequence"
] | [
"Existence of Sequence in Set of Real Numbers whose Limit is Infimum",
"Definition:Infimum of Set/Real Numbers",
"Definition:Distance/Sets/Metric Spaces"
] |
proofwiki-10224 | Limit of Sequence in Product of Metric Spaces under Chebyshev Distance | Let $M_1 = \struct {A_1, d_1}, M_2 = \struct {A_2, d_2}, \ldots, M_k = \struct {A_n, d_k}$ be metric spaces.
Let $\ds \AA = \prod_{i \mathop = 1}^k A_i$ be the cartesian product of $A_1, A_2, \ldots, A_k$.
Let $d_\infty: \AA \times \AA \to \R$ be the Chebyshev distance on $\AA$:
:$\ds \map {d_\infty} {x, y} = \max_{i \... | {{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} a_n
| r = c
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \lim_{n \mathop \to \infty} \map {d_\infty} {a_n, c}
| r = 0
| c = {{Defof|Convergent Sequence|subdef = Metric Space|index = 3}}
}}
{{eqn | ll= \leadstoandfrom
| l = \lim_{n ... | Let $M_1 = \struct {A_1, d_1}, M_2 = \struct {A_2, d_2}, \ldots, M_k = \struct {A_n, d_k}$ be [[Definition:Metric Space|metric spaces]].
Let $\ds \AA = \prod_{i \mathop = 1}^k A_i$ be the [[Definition:Finite Cartesian Product|cartesian product]] of $A_1, A_2, \ldots, A_k$.
Let $d_\infty: \AA \times \AA \to \R$ be the... | {{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} a_n
| r = c
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \lim_{n \mathop \to \infty} \map {d_\infty} {a_n, c}
| r = 0
| c = {{Defof|Convergent Sequence|subdef = Metric Space|index = 3}}
}}
{{eqn | ll= \leadstoandfrom
| l = \lim_{n ... | Limit of Sequence in Product of Metric Spaces under Chebyshev Distance | https://proofwiki.org/wiki/Limit_of_Sequence_in_Product_of_Metric_Spaces_under_Chebyshev_Distance | https://proofwiki.org/wiki/Limit_of_Sequence_in_Product_of_Metric_Spaces_under_Chebyshev_Distance | [
"Chebyshev Distance",
"Limits of Sequences"
] | [
"Definition:Metric Space",
"Definition:Cartesian Product/Finite",
"Definition:Chebyshev Distance",
"Definition:Sequence",
"Definition:Limit of Sequence/Metric Space"
] | [
"Limit of Image of Sequence"
] |
proofwiki-10225 | Recurrence Formula for Sum of Sequence of Fibonacci Numbers | Let $g_n$ be the sum of the first $n$ Fibonacci numbers.
Then:
:$\forall n \ge 2: g_n = g_{n - 1} + g_{n - 2} + 1$ | Let $F_n$ be the $n$th Fibonacci number.
By definition:
:$F_0 = 0, F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3, \ldots$
Hence, $g_n$, the sum of the first $n$ Fibonacci numbers is
:$g_0 = 0, g_1 = 1, g_2 = 2, g_3 = 4, g_4 = 7, \ldots$
Proof by induction:
For all $n \in \N_{>0}$, let $\map P n$ be the proposition:
:$\ds \sum_{j ... | Let $g_n$ be the sum of the first $n$ [[Definition:Fibonacci Numbers|Fibonacci numbers]].
Then:
:$\forall n \ge 2: g_n = g_{n - 1} + g_{n - 2} + 1$ | Let $F_n$ be the $n$th [[Definition:Fibonacci Numbers|Fibonacci number]].
By definition:
:$F_0 = 0, F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3, \ldots$
Hence, $g_n$, the sum of the first $n$ [[Definition:Fibonacci Numbers|Fibonacci numbers]] is
:$g_0 = 0, g_1 = 1, g_2 = 2, g_3 = 4, g_4 = 7, \ldots$
Proof by [[Principle of ... | Recurrence Formula for Sum of Sequence of Fibonacci Numbers | https://proofwiki.org/wiki/Recurrence_Formula_for_Sum_of_Sequence_of_Fibonacci_Numbers | https://proofwiki.org/wiki/Recurrence_Formula_for_Sum_of_Sequence_of_Fibonacci_Numbers | [
"Fibonacci Numbers",
"Recurrence Relations"
] | [
"Definition:Fibonacci Number"
] | [
"Definition:Fibonacci Number",
"Definition:Fibonacci Number",
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction",
"Definition:Fibonacci Number"
] |
proofwiki-10226 | Limit of Subsequence equals Limit of Sequence/Metric Space | Let $M = \struct {A, d}$ be a metric space.
Let $\sequence {x_n}$ be a sequence in $A$.
Let $l \in A$ such that:
:$\ds \lim_{n \mathop \to \infty} x_n = l$
Let $\sequence {x_{n_r} }$ be a subsequence of $\sequence {x_n}$.
Then:
:$\ds \lim_{r \mathop \to \infty} x_{n_r} = l$ | Let $\epsilon > 0$.
Since $\ds \lim_{n \mathop \to \infty} x_n = l$, it follows from the definition of limit that:
:$\exists N: \forall n > N: \map d {x_n, l} < \epsilon$
Now let $R = N$.
Then from Strictly Increasing Sequence of Natural Numbers:
:$\forall r > R: n_r \ge r$
Thus $n_r > N$ and so:
:$\map d {x_n, l} < \e... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $\sequence {x_n}$ be a [[Definition:Sequence|sequence in $A$]].
Let $l \in A$ such that:
:$\ds \lim_{n \mathop \to \infty} x_n = l$
Let $\sequence {x_{n_r} }$ be a [[Definition:Subsequence|subsequence]] of $\sequence {x_n}$.
Then:
:$\ds \l... | Let $\epsilon > 0$.
Since $\ds \lim_{n \mathop \to \infty} x_n = l$, it follows from the definition of [[Definition:Limit of Sequence (Metric Space)|limit]] that:
:$\exists N: \forall n > N: \map d {x_n, l} < \epsilon$
Now let $R = N$.
Then from [[Strictly Increasing Sequence of Natural Numbers]]:
:$\forall r > R: ... | Limit of Subsequence equals Limit of Sequence/Metric Space | https://proofwiki.org/wiki/Limit_of_Subsequence_equals_Limit_of_Sequence/Metric_Space | https://proofwiki.org/wiki/Limit_of_Subsequence_equals_Limit_of_Sequence/Metric_Space | [
"Metric Spaces",
"Convergence",
"Limits of Sequences"
] | [
"Definition:Metric Space",
"Definition:Sequence",
"Definition:Subsequence"
] | [
"Definition:Limit of Sequence/Metric Space",
"Strictly Increasing Sequence of Natural Numbers"
] |
proofwiki-10227 | Convergent Real Sequence is Bounded | Let $\sequence {x_n}$ be a sequence in $\R$.
Let $l \in A$ such that $\ds \lim_{n \mathop \to \infty} x_n = l$.
Then $\sequence {x_n}$ is bounded. | From Real Number Line is Metric Space, the set $\R$ under the usual metric is a metric space.
By Convergent Sequence in Metric Space is Bounded it follows that:
:$\exists M > 0: \forall n, m \in \N: \size {x_n - x_m} \le M$
Then for $n \in \N$, by the Triangle Inequality for Real Numbers:
{{begin-eqn}}
{{eqn | l = \siz... | Let $\sequence {x_n}$ be a [[Definition:Real Sequence|sequence in $\R$]].
Let $l \in A$ such that $\ds \lim_{n \mathop \to \infty} x_n = l$.
Then $\sequence {x_n}$ is [[Definition:Bounded Real Sequence|bounded]]. | From [[Real Number Line is Metric Space]], the set $\R$ under the [[Definition:Usual Metric|usual metric]] is a [[Definition:Metric Space|metric space]].
By [[Convergent Sequence in Metric Space is Bounded]] it follows that:
:$\exists M > 0: \forall n, m \in \N: \size {x_n - x_m} \le M$
Then for $n \in \N$, by the [[... | Convergent Real Sequence is Bounded/Proof 1 | https://proofwiki.org/wiki/Convergent_Real_Sequence_is_Bounded | https://proofwiki.org/wiki/Convergent_Real_Sequence_is_Bounded/Proof_1 | [
"Real Sequences",
"Limits of Sequences",
"Convergent Real Sequence is Bounded"
] | [
"Definition:Real Sequence",
"Definition:Bounded Sequence/Real"
] | [
"Real Number Line is Metric Space",
"Definition:Usual Metric",
"Definition:Metric Space",
"Convergent Sequence in Metric Space is Bounded",
"Triangle Inequality/Real Numbers",
"Definition:Bounded Sequence/Real"
] |
proofwiki-10228 | Convergent Real Sequence is Bounded | Let $\sequence {x_n}$ be a sequence in $\R$.
Let $l \in A$ such that $\ds \lim_{n \mathop \to \infty} x_n = l$.
Then $\sequence {x_n}$ is bounded. | Let $\sequence {x_n}$ be a sequence in $\R$.
Let $x_n \to l$ as $n \to \infty$.
To show that $\sequence {x_n}$ is bounded sequence, we need to find $K$ such that:
:$\forall n \in \N: \size {x_n} \le K$
Because $\sequence {x_n}$ converges:
:$\forall \epsilon > 0: \exists N: n > N \implies \size {x_n - l} < \epsilon$
In ... | Let $\sequence {x_n}$ be a [[Definition:Real Sequence|sequence in $\R$]].
Let $l \in A$ such that $\ds \lim_{n \mathop \to \infty} x_n = l$.
Then $\sequence {x_n}$ is [[Definition:Bounded Real Sequence|bounded]]. | Let $\sequence {x_n}$ be a [[Definition:Real Sequence|sequence in $\R$]].
Let $x_n \to l$ as $n \to \infty$.
To show that $\sequence {x_n}$ is [[Definition:Bounded Sequence|bounded sequence]], we need to find $K$ such that:
:$\forall n \in \N: \size {x_n} \le K$
Because $\sequence {x_n}$ [[Definition:Convergent Real... | Convergent Real Sequence is Bounded/Proof 2 | https://proofwiki.org/wiki/Convergent_Real_Sequence_is_Bounded | https://proofwiki.org/wiki/Convergent_Real_Sequence_is_Bounded/Proof_2 | [
"Real Sequences",
"Limits of Sequences",
"Convergent Real Sequence is Bounded"
] | [
"Definition:Real Sequence",
"Definition:Bounded Sequence/Real"
] | [
"Definition:Real Sequence",
"Definition:Bounded Sequence",
"Definition:Convergent Sequence/Real Numbers",
"Reverse Triangle Inequality/Real and Complex Fields/Corollary 1"
] |
proofwiki-10229 | Monotone Convergence Theorem (Real Analysis)/Decreasing Sequence | Let $\sequence {x_n}$ be a decreasing real sequence which is bounded below.
Then $\sequence {x_n}$ converges to its infimum. | Let $\sequence {x_n}$ be decreasing and bounded below.
Then $\sequence {-x_n}$ is increasing and bounded above.
Thus the Monotone Convergence Theorem for Increasing Sequence applies and the proof follows.
{{Qed}} | Let $\sequence {x_n}$ be a [[Definition:Decreasing Real Sequence|decreasing real sequence]] which is [[Definition:Bounded Below Real Sequence|bounded below]].
Then $\sequence {x_n}$ [[Definition:Convergent Real Sequence|converges]] to its [[Definition:Infimum of Sequence|infimum]]. | Let $\sequence {x_n}$ be [[Definition:Decreasing Real Sequence|decreasing]] and [[Definition:Bounded Below Real Sequence|bounded below]].
Then $\sequence {-x_n}$ is [[Definition:Increasing Real Sequence|increasing]] and [[Definition:Bounded Above Real Sequence|bounded above]].
Thus the [[Monotone Convergence Theorem ... | Monotone Convergence Theorem (Real Analysis)/Decreasing Sequence | https://proofwiki.org/wiki/Monotone_Convergence_Theorem_(Real_Analysis)/Decreasing_Sequence | https://proofwiki.org/wiki/Monotone_Convergence_Theorem_(Real_Analysis)/Decreasing_Sequence | [
"Monotone Convergence Theorem (Real Analysis)"
] | [
"Definition:Decreasing/Sequence/Real Sequence",
"Definition:Bounded Below Sequence/Real",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Infimum of Sequence"
] | [
"Definition:Decreasing/Sequence/Real Sequence",
"Definition:Bounded Below Sequence/Real",
"Definition:Increasing/Sequence/Real Sequence",
"Definition:Bounded Above Sequence/Real",
"Monotone Convergence Theorem (Real Analysis)/Increasing Sequence"
] |
proofwiki-10230 | Monotone Convergence Theorem (Real Analysis)/Increasing Sequence | Let $\sequence {x_n}$ be an increasing real sequence which is bounded above.
Then $\sequence {x_n}$ converges to its supremum. | Suppose $\sequence {x_n}$ is increasing and bounded above.
By the Continuum Property, it has a supremum, $B$.
We need to show that $x_n \to B$ as $n \to \infty$.
Let $\epsilon \in \R_{>0}$.
By the definition of supremum, $B - \epsilon$ is not an upper bound.
Thus:
:$\exists N \in \N: x_N > B - \epsilon$
But $\sequence ... | Let $\sequence {x_n}$ be an [[Definition:Increasing Real Sequence|increasing real sequence]] which is [[Definition:Bounded Above Real Sequence|bounded above]].
Then $\sequence {x_n}$ [[Definition:Convergent Real Sequence|converges]] to its [[Definition:Supremum of Sequence|supremum]]. | Suppose $\sequence {x_n}$ is [[Definition:Increasing Real Sequence|increasing]] and [[Definition:Bounded Above Real Sequence|bounded above]].
By the [[Continuum Property]], it has a [[Definition:Supremum of Sequence|supremum]], $B$.
We need to show that $x_n \to B$ as $n \to \infty$.
Let $\epsilon \in \R_{>0}$.
By ... | Monotone Convergence Theorem (Real Analysis)/Increasing Sequence | https://proofwiki.org/wiki/Monotone_Convergence_Theorem_(Real_Analysis)/Increasing_Sequence | https://proofwiki.org/wiki/Monotone_Convergence_Theorem_(Real_Analysis)/Increasing_Sequence | [
"Monotone Convergence Theorem (Real Analysis)"
] | [
"Definition:Increasing/Sequence/Real Sequence",
"Definition:Bounded Above Sequence/Real",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Supremum of Sequence"
] | [
"Definition:Increasing/Sequence/Real Sequence",
"Definition:Bounded Above Sequence/Real",
"Continuum Property",
"Definition:Supremum of Sequence",
"Definition:Supremum of Sequence",
"Definition:Upper Bound of Sequence",
"Definition:Increasing/Sequence",
"Definition:Upper Bound of Sequence",
"Real Pl... |
proofwiki-10231 | Open Ball in Euclidean Plane is Interior of Circle | Let $\R^2$ be the real number plane with the usual (Euclidean) metric.
Let $x = \tuple {x_1, x_2} \in \R^2$ be a point in $\R^2$.
Let $\map {B_\epsilon} x$ be the open $\epsilon$-ball at $x$.
Then $\map {B_\epsilon} x$ is the interior of the circle whose center is $x$ and whose radius is $\epsilon$. | Let $S = \map {B_\epsilon} x$ be an open $\epsilon$-ball at $x$.
Let $y = \tuple {y_1, y_2} \in \map {B_\epsilon} x$.
Then:
{{begin-eqn}}
{{eqn | l = y
| o = \in
| m = \map {B_\epsilon} x
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \map d {y, x}
| o = <
| m = \epsilon
| c = {{D... | Let $\R^2$ be the [[Definition:Real Number Plane with Euclidean Metric|real number plane with the usual (Euclidean) metric]].
Let $x = \tuple {x_1, x_2} \in \R^2$ be a point in $\R^2$.
Let $\map {B_\epsilon} x$ be the [[Definition:Open Ball of Metric Space|open $\epsilon$-ball at $x$]].
Then $\map {B_\epsilon} x$ i... | Let $S = \map {B_\epsilon} x$ be an [[Definition:Open Ball of Metric Space|open $\epsilon$-ball at $x$]].
Let $y = \tuple {y_1, y_2} \in \map {B_\epsilon} x$.
Then:
{{begin-eqn}}
{{eqn | l = y
| o = \in
| m = \map {B_\epsilon} x
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \map d {y, x}
|... | Open Ball in Euclidean Plane is Interior of Circle | https://proofwiki.org/wiki/Open_Ball_in_Euclidean_Plane_is_Interior_of_Circle | https://proofwiki.org/wiki/Open_Ball_in_Euclidean_Plane_is_Interior_of_Circle | [
"Open Balls",
"Real Number Plane with Euclidean Metric"
] | [
"Definition:Euclidean Metric/Real Number Plane",
"Definition:Open Ball",
"Definition:Region",
"Definition:Circle",
"Definition:Circle/Center",
"Definition:Circle/Radius"
] | [
"Definition:Open Ball",
"Equation of Circle",
"Definition:Circle",
"Definition:Circle/Center",
"Definition:Circle/Radius",
"Definition:Region",
"Open Ball of Point Inside Open Ball"
] |
proofwiki-10232 | Open Ball in Euclidean 3-Space is Interior of Sphere | Let $\R^3$ be the real Euclidean $3$-space considered as a metric space under the usual metric.
Let $x = \tuple {x_1, x_2, x_3} \in \R^3$ be a point in $\R^3$.
Let $\map {B_\epsilon} x$ be the open $\epsilon$-ball at $x$.
Then $\map {B_\epsilon} x$ is the interior of the sphere whose center is $x$ and whose radius is $... | Let $S = \map {B_\epsilon} x$ be an open $\epsilon$-ball at $x$.
Let $y = \tuple {y_1, y_2, y_3} \in \map {B_\epsilon} x$.
Then:
{{begin-eqn}}
{{eqn | l = y
| o = \in
| m = \map {B_\epsilon} x
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \map d {y, x}
| o = <
| m = \epsilon
| c ... | Let $\R^3$ be the [[Definition:Real Euclidean Space|real Euclidean $3$-space]] considered as a [[Definition:Metric Space|metric space]] under the [[Definition:Usual Metric|usual metric]].
Let $x = \tuple {x_1, x_2, x_3} \in \R^3$ be a point in $\R^3$.
Let $\map {B_\epsilon} x$ be the [[Definition:Open Ball of Metric ... | Let $S = \map {B_\epsilon} x$ be an [[Definition:Open Ball of Metric Space|open $\epsilon$-ball at $x$]].
Let $y = \tuple {y_1, y_2, y_3} \in \map {B_\epsilon} x$.
Then:
{{begin-eqn}}
{{eqn | l = y
| o = \in
| m = \map {B_\epsilon} x
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \map d {y, x}
... | Open Ball in Euclidean 3-Space is Interior of Sphere | https://proofwiki.org/wiki/Open_Ball_in_Euclidean_3-Space_is_Interior_of_Sphere | https://proofwiki.org/wiki/Open_Ball_in_Euclidean_3-Space_is_Interior_of_Sphere | [
"Open Balls",
"Real Euclidean Spaces"
] | [
"Definition:Euclidean Space/Real",
"Definition:Metric Space",
"Definition:Usual Metric",
"Definition:Open Ball",
"Definition:Region",
"Definition:Sphere/Geometry",
"Definition:Sphere/Geometry/Center",
"Definition:Sphere/Geometry/Radius"
] | [
"Definition:Open Ball",
"Equation of Sphere/Rectangular Coordinates",
"Definition:Sphere/Geometry",
"Definition:Sphere/Geometry/Center",
"Definition:Sphere/Geometry/Radius",
"Definition:Region",
"Open Ball of Point Inside Open Ball"
] |
proofwiki-10233 | Open Ball in Real Number Plane under Chebyshev Distance | Let $\R^2$ be the real number plane.
Let $d_\infty: \R^2 \times \R^2 \to \R$ be the Chebyshev Distance on $\R^2$:
:$\ds \map {d_\infty} {x, y} := \max \set {\size {x_1 - y_1}, \size {x_2 - y_2} }$
where $x = \tuple {x_1, x_2}, y = \tuple {y_1, y_2} \in \R^2$.
For $a \in \R^2$, let $\map {B_\epsilon} a$ be the open $\ep... | Let $a = \tuple {a_1, a_2}$.
From Open Ball in Cartesian Product under Chebyshev Distance:
:$\map {B_\epsilon} {a; d_\infty} = \map {B_\epsilon} {a_1; d} \times \map {B_\epsilon} {a_2; d}$
where $d$ is the usual (Euclidean) topology.
From Open Ball in Real Number Line is Open Interval:
:$\map {B_\epsilon} {a_1; d} \tim... | Let $\R^2$ be the [[Definition:Real Number Plane|real number plane]].
Let $d_\infty: \R^2 \times \R^2 \to \R$ be the [[Definition:Chebyshev Distance|Chebyshev Distance]] on $\R^2$:
:$\ds \map {d_\infty} {x, y} := \max \set {\size {x_1 - y_1}, \size {x_2 - y_2} }$
where $x = \tuple {x_1, x_2}, y = \tuple {y_1, y_2} \... | Let $a = \tuple {a_1, a_2}$.
From [[Open Ball in Cartesian Product under Chebyshev Distance]]:
:$\map {B_\epsilon} {a; d_\infty} = \map {B_\epsilon} {a_1; d} \times \map {B_\epsilon} {a_2; d}$
where $d$ is the [[Definition:Euclidean Topology on Real Number Line|usual (Euclidean) topology]].
From [[Open Ball in Real N... | Open Ball in Real Number Plane under Chebyshev Distance | https://proofwiki.org/wiki/Open_Ball_in_Real_Number_Plane_under_Chebyshev_Distance | https://proofwiki.org/wiki/Open_Ball_in_Real_Number_Plane_under_Chebyshev_Distance | [
"Open Balls",
"Chebyshev Distance"
] | [
"Definition:Real Number Plane",
"Definition:Chebyshev Distance",
"Definition:Open Ball",
"Definition:Region",
"Definition:Quadrilateral/Square",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Axis/Coordinate Axes"
] | [
"Open Ball in Cartesian Product under Chebyshev Distance",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Open Ball in Real Number Line is Open Interval"
] |
proofwiki-10234 | Open Ball in Standard Discrete Metric Space | Let $M = \struct {A, d}$ be a metric space.
Let $d$ be the standard discrete metric on $M$.
Let $a \in A$.
Let $\map {B_\epsilon} {a; d}$ be an open $\epsilon$-ball of $a$ in $M$.
Then:
:$\map {B_\epsilon} {a; d} = \begin {cases} \set a & : \epsilon \le 1 \\ A & : \epsilon > 1 \end {cases}$ | Let $\epsilon \in \R_{>0}: \epsilon \le 1$.
Then:
{{begin-eqn}}
{{eqn | q = \forall x \in A
| l = x
| o = \ne
| r = a
| c =
}}
{{eqn | ll= \leadsto
| l = \map d {x, a}
| o = \ge
| r = \epsilon
| c = {{Defof|Standard Discrete Metric}}
}}
{{eqn | ll= \leadsto
| l = x... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $d$ be the [[Definition:Standard Discrete Metric|standard discrete metric]] on $M$.
Let $a \in A$.
Let $\map {B_\epsilon} {a; d}$ be an [[Definition:Open Ball of Metric Space|open $\epsilon$-ball]] of $a$ in $M$.
Then:
:$\map {B_\epsilon}... | Let $\epsilon \in \R_{>0}: \epsilon \le 1$.
Then:
{{begin-eqn}}
{{eqn | q = \forall x \in A
| l = x
| o = \ne
| r = a
| c =
}}
{{eqn | ll= \leadsto
| l = \map d {x, a}
| o = \ge
| r = \epsilon
| c = {{Defof|Standard Discrete Metric}}
}}
{{eqn | ll= \leadsto
| l =... | Open Ball in Standard Discrete Metric Space | https://proofwiki.org/wiki/Open_Ball_in_Standard_Discrete_Metric_Space | https://proofwiki.org/wiki/Open_Ball_in_Standard_Discrete_Metric_Space | [
"Open Balls",
"Standard Discrete Metric"
] | [
"Definition:Metric Space",
"Definition:Standard Discrete Metric",
"Definition:Open Ball"
] | [] |
proofwiki-10235 | Distance from Point to Subset is Continuous Function | Let $M = \struct {X, d}$ be a metric space.
Let $A \subseteq X$ be a non-empty subset of $X$.
Let $f: X \to \R$ be the function defined as:
:$\forall x \in X: \map f x = \map d {x, A}$
where $\map d {x, A}$ denotes the distance from $x$ to $A$.
Then $f$ is continuous. | $\forall x, y \in X, \forall z \in A$, by the definition of the distance from $x$ to $A$, we have:
:$\map d {x, A} \le \map d {x, z} \le \map d {x, y} + \map d {y, z}$
From Triangle Inequality on Distance from Point to Subset:
:$\map d {x, A} \le \map d {x, y} + \map d {y, A}$
and:
:$\map d {y, A} \le \map d {x, y} + \... | Let $M = \struct {X, d}$ be a [[Definition:Metric Space|metric space]].
Let $A \subseteq X$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of $X$.
Let $f: X \to \R$ be the [[Definition:Real-Valued Function|function]] defined as:
:$\forall x \in X: \map f x = \map d {x, A}$
where $\map d {x,... | $\forall x, y \in X, \forall z \in A$, by the definition of the [[Definition:Distance between Element and Subset of Metric Space|distance from $x$ to $A$]], we have:
:$\map d {x, A} \le \map d {x, z} \le \map d {x, y} + \map d {y, z}$
From [[Triangle Inequality on Distance from Point to Subset]]:
:$\map d {x, A} \le \... | Distance from Point to Subset is Continuous Function | https://proofwiki.org/wiki/Distance_from_Point_to_Subset_is_Continuous_Function | https://proofwiki.org/wiki/Distance_from_Point_to_Subset_is_Continuous_Function | [
"Continuous Mappings",
"Distance Function"
] | [
"Definition:Metric Space",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Real-Valued Function",
"Definition:Distance/Sets/Metric Spaces",
"Definition:Continuous Mapping (Metric Space)"
] | [
"Definition:Distance/Sets/Metric Spaces",
"Triangle Inequality on Distance from Point to Subset",
"Definition:Continuous Mapping (Metric Space)"
] |
proofwiki-10236 | Neighbourhood of Point Contains Point of Subset iff Distance is Zero | Let $M = \struct {X, d}$ be a metric space.
Let $A \subseteq X$ be a non-empty subset of $X$.
Let $x \in X$.
Then every neighborhood of $x$ contains a point of $A$ {{iff}}:
:$\map d {x, A} = 0$
where $\map d {x, A}$ denotes the distance from $x$ to $A$. | === Sufficient Condition ===
Let $x \in X$.
Let every neighborhood of $x$ contain a point in $A$.
Every open ball $\map {B_\epsilon} x$ centered at $x$ is seen to be a neighborhood of $x$ in $M$.
But then this implies that for every $\epsilon \in \R_{\gt 0}$ there must exists a $y \in A$ such that:
:$\map d {x, y} < \e... | Let $M = \struct {X, d}$ be a [[Definition:Metric Space|metric space]].
Let $A \subseteq X$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of $X$.
Let $x \in X$.
Then every [[Definition:Neighborhood (Metric Space)|neighborhood]] of $x$ contains a point of $A$ {{iff}}:
:$\map d {x, A} = 0$
... | === Sufficient Condition ===
Let $x \in X$.
Let every [[Definition:Neighborhood (Metric Space)|neighborhood]] of $x$ contain a point in $A$.
Every [[Definition:Open Ball|open ball]] $\map {B_\epsilon} x$ [[Definition:Center of Open Ball|centered]] at $x$ is seen to be a [[Definition:Neighborhood (Metric Space)|neigh... | Neighbourhood of Point Contains Point of Subset iff Distance is Zero | https://proofwiki.org/wiki/Neighbourhood_of_Point_Contains_Point_of_Subset_iff_Distance_is_Zero | https://proofwiki.org/wiki/Neighbourhood_of_Point_Contains_Point_of_Subset_iff_Distance_is_Zero | [
"Neighborhoods",
"Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Neighborhood (Metric Space)",
"Definition:Distance/Sets/Metric Spaces"
] | [
"Definition:Neighborhood (Metric Space)",
"Definition:Open Ball",
"Definition:Open Ball/Center",
"Definition:Neighborhood (Metric Space)",
"Definition:Neighborhood (Metric Space)",
"Definition:Neighborhood (Metric Space)",
"Definition:Open Ball",
"Definition:Open Ball/Center"
] |
proofwiki-10237 | Mapping from Cartesian Product under Chebyshev Distance to Real Number Line is Continuous | Let $M = \struct {A, d'}$ be a metric space.
Let $\ds \AA = A \times A$ be the cartesian product of $A$ with itself.
Let $d_\infty: \AA \times \AA \to \R$ be the Chebyshev distance on $\AA$:
:$\ds \map {d_\infty} {x, y} = \max \set {\map {d'} {x_1, y_1}, \map {d'} {x_2, y_2} }$
where $x = \tuple {x_1, x_2}, y = \tuple ... | From definition of continuous mapping:
We just need to show that:
:$\forall \tuple {a, b} \in A \times A: \forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall \tuple {x, y} \in A \times A: \map {d_\infty} {\tuple {x, y}, \tuple {a, b}} < \delta \implies \map d {\map {d'} {x, y}, \map {d'} {a, b}} < \epsilo... | Let $M = \struct {A, d'}$ be a [[Definition:Metric Space|metric space]].
Let $\ds \AA = A \times A$ be the [[Definition:Cartesian Product|cartesian product]] of $A$ with itself.
Let $d_\infty: \AA \times \AA \to \R$ be the [[Definition:Chebyshev Distance|Chebyshev distance]] on $\AA$:
:$\ds \map {d_\infty} {x, y} = \... | From definition of [[Definition:Continuous Mapping (Metric Space)|continuous mapping]]:
We just need to show that:
:$\forall \tuple {a, b} \in A \times A: \forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall \tuple {x, y} \in A \times A: \map {d_\infty} {\tuple {x, y}, \tuple {a, b}} < \delta \implies \ma... | Mapping from Cartesian Product under Chebyshev Distance to Real Number Line is Continuous | https://proofwiki.org/wiki/Mapping_from_Cartesian_Product_under_Chebyshev_Distance_to_Real_Number_Line_is_Continuous | https://proofwiki.org/wiki/Mapping_from_Cartesian_Product_under_Chebyshev_Distance_to_Real_Number_Line_is_Continuous | [
"Continuous Mappings",
"Chebyshev Distance"
] | [
"Definition:Metric Space",
"Definition:Cartesian Product",
"Definition:Chebyshev Distance",
"Definition:Mapping",
"Definition:Usual Metric",
"Definition:Continuous Mapping (Metric Space)"
] | [
"Definition:Continuous Mapping (Metric Space)",
"Triangle Inequality",
"Reverse Triangle Inequality",
"Definition:Continuous Mapping (Metric Space)"
] |
proofwiki-10238 | Empty Set is Open and Closed in Metric Space | Let $M = \struct {A, d}$ be a metric space.
Then the empty set $\O$ is both open and closed in $M$. | From Empty Set is Open in Metric Space, $\O$ is open in $M$.
From Empty Set is Closed in Metric Space, $\O$ is closed in $M$.
{{qed}} | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Then the [[Definition:Empty Set|empty set]] $\O$ is both [[Definition:Open Set (Metric Space)|open]] and [[Definition:Closed Set (Metric Space)|closed]] in $M$. | From [[Empty Set is Open in Metric Space]], $\O$ is [[Definition:Open Set (Metric Space)|open]] in $M$.
From [[Empty Set is Closed in Metric Space]], $\O$ is [[Definition:Closed Set (Metric Space)|closed]] in $M$.
{{qed}} | Empty Set is Open and Closed in Metric Space | https://proofwiki.org/wiki/Empty_Set_is_Open_and_Closed_in_Metric_Space | https://proofwiki.org/wiki/Empty_Set_is_Open_and_Closed_in_Metric_Space | [
"Open Sets (Metric Spaces)",
"Closed Sets",
"Empty Set",
"Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Empty Set",
"Definition:Open Set/Metric Space",
"Definition:Closed Set/Metric Space"
] | [
"Empty Set is Open in Metric Space",
"Definition:Open Set/Metric Space",
"Empty Set is Closed/Metric Space",
"Definition:Closed Set/Metric Space"
] |
proofwiki-10239 | Empty Set is Closed/Metric Space | Let $M = \struct {A, d}$ be a metric space.
Then the empty set $\O$ is closed in $M$. | From Metric Space is Open in Itself, $A$ is open in $M$.
But:
:$\O = \relcomp A A$
where $\complement_A$ denotes the set complement relative to $A$.
The result follows by definition of closed set.
{{qed}} | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Then the [[Definition:Empty Set|empty set]] $\O$ is [[Definition:Closed Set (Metric Space)|closed]] in $M$. | From [[Metric Space is Open in Itself]], $A$ is [[Definition:Open Set (Metric Space)|open]] in $M$.
But:
:$\O = \relcomp A A$
where $\complement_A$ denotes the [[Definition:Relative Complement|set complement relative to $A$]].
The result follows by definition of [[Definition:Closed Set/Metric Space/Definition 1|close... | Empty Set is Closed/Metric Space | https://proofwiki.org/wiki/Empty_Set_is_Closed/Metric_Space | https://proofwiki.org/wiki/Empty_Set_is_Closed/Metric_Space | [
"Open Sets (Metric Spaces)",
"Closed Sets (Metric Spaces)",
"Closed Sets (Metric Spaces)",
"Empty Set is Closed"
] | [
"Definition:Metric Space",
"Definition:Empty Set",
"Definition:Closed Set/Metric Space"
] | [
"Metric Space is Open in Itself",
"Definition:Open Set/Metric Space",
"Definition:Relative Complement",
"Definition:Closed Set/Metric Space/Definition 1"
] |
proofwiki-10240 | Metric Space is Closed in Itself | Let $M = \struct {A, d}$ be a metric space.
Then $A$ is closed in $M$. | From Empty Set is Open in Metric Space, $\O$ is open in $M$.
But:
:$A = \relcomp A \O$
where $\complement_A$ denotes the set complement relative to $A$.
The result follows by definition of closed set.
{{qed}} | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Then $A$ is [[Definition:Closed Set (Metric Space)|closed]] in $M$. | From [[Empty Set is Open in Metric Space]], $\O$ is [[Definition:Open Set (Metric Space)|open]] in $M$.
But:
:$A = \relcomp A \O$
where $\complement_A$ denotes the [[Definition:Relative Complement|set complement relative to $A$]].
The result follows by definition of [[Definition:Closed Set/Metric Space/Definition 1|c... | Metric Space is Closed in Itself | https://proofwiki.org/wiki/Metric_Space_is_Closed_in_Itself | https://proofwiki.org/wiki/Metric_Space_is_Closed_in_Itself | [
"Metric Spaces",
"Closed Sets"
] | [
"Definition:Metric Space",
"Definition:Closed Set/Metric Space"
] | [
"Empty Set is Open in Metric Space",
"Definition:Open Set/Metric Space",
"Definition:Relative Complement",
"Definition:Closed Set/Metric Space/Definition 1"
] |
proofwiki-10241 | Metric Space is Open and Closed in Itself | Let $M = \struct {A, d}$ be a metric space.
Then $A$ is both open and closed in $M$. | From Metric Space is Open in Itself, $A$ is open in $M$.
From Metric Space is Closed in Itself, $A$ is closed in $M$.
{{qed}} | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Then $A$ is both [[Definition:Open Set (Metric Space)|open]] and [[Definition:Closed Set (Metric Space)|closed]] in $M$. | From [[Metric Space is Open in Itself]], $A$ is [[Definition:Open Set (Metric Space)|open]] in $M$.
From [[Metric Space is Closed in Itself]], $A$ is [[Definition:Closed Set (Metric Space)|closed]] in $M$.
{{qed}} | Metric Space is Open and Closed in Itself | https://proofwiki.org/wiki/Metric_Space_is_Open_and_Closed_in_Itself | https://proofwiki.org/wiki/Metric_Space_is_Open_and_Closed_in_Itself | [
"Metric Spaces",
"Open Sets (Metric Spaces)",
"Closed Sets (Metric Spaces)"
] | [
"Definition:Metric Space",
"Definition:Open Set/Metric Space",
"Definition:Closed Set/Metric Space"
] | [
"Metric Space is Open in Itself",
"Definition:Open Set/Metric Space",
"Metric Space is Closed in Itself",
"Definition:Closed Set/Metric Space"
] |
proofwiki-10242 | Subset of Metric Space contains Limits of Sequences iff Closed | Let $M = \struct {A, d}$ be a metric space.
Let $H \subseteq A$.
Then $H$ is closed in $M$ {{iff}}:
:for each sequence $\sequence {a_n}$ of points of $H$ that converges to a point $a \in A$, it follows that $a \in H$. | === Necessary Condition ===
Let $H$ be closed in $M$.
Suppose that:
:$\ds \lim_{n \mathop \to \infty} a_n = a$
and:
:$\forall n \in \N_{>0}: a_n \in H$
If the set $\set {a_1, a_2, \ldots}$ is infinite then every neighborhood of $a$ contains infinitely many points of $H$.
Thus $a$ is a limit point of $H$.
So by definiti... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $H \subseteq A$.
Then $H$ is [[Definition:Closed Set (Metric Space)|closed]] in $M$ {{iff}}:
:for each [[Definition:Sequence|sequence]] $\sequence {a_n}$ of points of $H$ that [[Definition:Convergent Sequence (Metric Space)|converges]] to a ... | === Necessary Condition ===
Let $H$ be [[Definition:Closed Set (Metric Space)|closed]] in $M$.
Suppose that:
:$\ds \lim_{n \mathop \to \infty} a_n = a$
and:
:$\forall n \in \N_{>0}: a_n \in H$
If the set $\set {a_1, a_2, \ldots}$ is [[Definition:Infinite Set|infinite]] then every [[Definition:Neighborhood (Metric Sp... | Subset of Metric Space contains Limits of Sequences iff Closed | https://proofwiki.org/wiki/Subset_of_Metric_Space_contains_Limits_of_Sequences_iff_Closed | https://proofwiki.org/wiki/Subset_of_Metric_Space_contains_Limits_of_Sequences_iff_Closed | [
"Metric Spaces",
"Closed Sets"
] | [
"Definition:Metric Space",
"Definition:Closed Set/Metric Space",
"Definition:Sequence",
"Definition:Convergent Sequence/Metric Space"
] | [
"Definition:Closed Set/Metric Space",
"Definition:Infinite Set",
"Definition:Neighborhood (Metric Space)",
"Definition:Limit Point/Metric Space",
"Definition:Closed Set/Metric Space/Definition 2",
"Definition:Finite Set",
"Definition:Limit Point/Metric Space",
"Definition:Closed Set/Metric Space/Defin... |
proofwiki-10243 | Subset of Metric Space is Closed iff contains all Zero Distance Points | Let $M = \struct {A, d}$ be a metric space.
Let $H \subseteq A$.
Then $H$ is closed in $M$ {{iff}}:
:$\forall x \in A: \map d {x, H} = 0 \implies x \in H$
where $\map d {x, H}$ denotes the distance between $x$ and $H$. | === Necessary Condition ===
Let $H$ be closed in $M$.
Let $x \in A: \map d {x, H} = 0 \implies x \in H$.
By Existence of Sequence in Subset of Metric Space whose Limit is Infimum there exists a sequence $\sequence {a_n}$ of points of $H$ such that:
:$\ds \lim_{n \mathop \to \infty} \map d {x, a_n} = 0$
So every neighbo... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $H \subseteq A$.
Then $H$ is [[Definition:Closed Set (Metric Space)|closed]] in $M$ {{iff}}:
:$\forall x \in A: \map d {x, H} = 0 \implies x \in H$
where $\map d {x, H}$ denotes the [[Definition:Distance between Element and Subset of Metric ... | === Necessary Condition ===
Let $H$ be [[Definition:Closed Set (Metric Space)|closed]] in $M$.
Let $x \in A: \map d {x, H} = 0 \implies x \in H$.
By [[Existence of Sequence in Subset of Metric Space whose Limit is Infimum]] there exists a [[Definition:Sequence|sequence]] $\sequence {a_n}$ of points of $H$ such that:... | Subset of Metric Space is Closed iff contains all Zero Distance Points | https://proofwiki.org/wiki/Subset_of_Metric_Space_is_Closed_iff_contains_all_Zero_Distance_Points | https://proofwiki.org/wiki/Subset_of_Metric_Space_is_Closed_iff_contains_all_Zero_Distance_Points | [
"Metric Spaces",
"Closed Sets"
] | [
"Definition:Metric Space",
"Definition:Closed Set/Metric Space",
"Definition:Distance/Sets/Metric Spaces"
] | [
"Definition:Closed Set/Metric Space",
"Existence of Sequence in Subset of Metric Space whose Limit is Infimum",
"Definition:Sequence",
"Definition:Neighborhood (Metric Space)",
"Definition:Limit of Sequence/Metric Space",
"Definition:Closed Set/Metric Space/Definition 2",
"Definition:Closed Set/Metric S... |
proofwiki-10244 | Continuity of Mapping between Metric Spaces by Closed Sets | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.
Let $f: A_1 \to A_2$ be a mapping.
Then $f$ is continuous {{iff}}:
:for every $V \subseteq A_2$ which is closed in $M_2$, $f^{-1} \sqbrk V$ is closed in $M_1$. | First the following is established.
Let $W \in A_2$.
We note that:
:$f^{-1} \sqbrk {A_2} = A_1$
Hence, from Preimage of Set Difference under Mapping:
:$f^{-1} \sqbrk {A_2 \setminus W} = A_1 \setminus f^{-1} \sqbrk W$ | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be [[Definition:Metric Space|metric spaces]].
Let $f: A_1 \to A_2$ be a [[Definition:Mapping|mapping]].
Then $f$ is [[Definition:Continuous Mapping (Metric Spaces)|continuous]] {{iff}}:
:for every $V \subseteq A_2$ which is [[Definition:Closed Set (Metric... | First the following is established.
Let $W \in A_2$.
We note that:
:$f^{-1} \sqbrk {A_2} = A_1$
Hence, from [[Preimage of Set Difference under Mapping]]:
:$f^{-1} \sqbrk {A_2 \setminus W} = A_1 \setminus f^{-1} \sqbrk W$ | Continuity of Mapping between Metric Spaces by Closed Sets | https://proofwiki.org/wiki/Continuity_of_Mapping_between_Metric_Spaces_by_Closed_Sets | https://proofwiki.org/wiki/Continuity_of_Mapping_between_Metric_Spaces_by_Closed_Sets | [
"Metric Spaces",
"Closed Sets",
"Continuous Mappings"
] | [
"Definition:Metric Space",
"Definition:Mapping",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Closed Set/Metric Space",
"Definition:Closed Set/Metric Space"
] | [
"Preimage of Set Difference under Mapping"
] |
proofwiki-10245 | Metric Space defined by Closed Sets | Let $M = \struct {A, d}$ be a metric space.
Then:
{{begin-axiom}}
{{axiom | n = \text C 1
| lc= $A$ is closed in $M$
}}
{{axiom | n = \text C 2
| lc= $\O$ is closed in $M$
}}
{{axiom | n = \text C 3
| lc= The union of a finite number of closed sets of $M$ is a closed set of $M$
}}
{{axiom | n = ... | From Metric Space is Closed in Itself, $\text C 1$ holds.
{{qed|lemma}}
From Empty Set is Closed in Metric Space, $\text C 2$ holds.
{{qed|lemma}}
Let $\ds \bigcup_{i \mathop = 1}^n V_i$ be the union of a finite number of closed sets of $M$.
Then from De Morgan's laws:
:$\ds A \setminus \bigcup_{i \mathop = 1}^n V_i = ... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Then:
{{begin-axiom}}
{{axiom | n = \text C 1
| lc= $A$ is [[Definition:Closed Set (Metric Space)|closed]] in $M$
}}
{{axiom | n = \text C 2
| lc= $\O$ is [[Definition:Closed Set (Metric Space)|closed]] in $M$
}}
{{axiom | n = \tex... | From [[Metric Space is Closed in Itself]], $\text C 1$ holds.
{{qed|lemma}}
From [[Empty Set is Closed in Metric Space]], $\text C 2$ holds.
{{qed|lemma}}
Let $\ds \bigcup_{i \mathop = 1}^n V_i$ be the [[Definition:Set Union|union]] of a [[Definition:Finite|finite]] number of [[Definition:Closed Set (Metric Space)|... | Metric Space defined by Closed Sets | https://proofwiki.org/wiki/Metric_Space_defined_by_Closed_Sets | https://proofwiki.org/wiki/Metric_Space_defined_by_Closed_Sets | [
"Metric Spaces",
"Closed Sets"
] | [
"Definition:Metric Space",
"Definition:Closed Set/Metric Space",
"Definition:Closed Set/Metric Space",
"Definition:Set Union",
"Definition:Finite Set",
"Definition:Closed Set/Metric Space",
"Definition:Closed Set/Metric Space",
"Definition:Set Intersection",
"Definition:Closed Set/Metric Space",
"... | [
"Metric Space is Closed in Itself",
"Empty Set is Closed/Metric Space",
"Definition:Set Union",
"Definition:Finite",
"Definition:Closed Set/Metric Space",
"De Morgan's Laws (Set Theory)",
"Definition:Closed Set/Metric Space",
"Definition:Open Set/Metric Space",
"Definition:Set Intersection",
"Defi... |
proofwiki-10246 | Infinite Union of Closed Sets of Metric Space may not be Closed | Let $M = \struct {A, d}$ be a metric space.
Let $V_1, V_2, V_3, \ldots$ be an infinite set of closed sets of $M$.
Then it is not necessarily the case that $\ds \bigcup_{n \mathop \in \N} V_n$ is itself a closed set of $M$. | Consider the closed real interval:
:$\closedint {\dfrac 1 n} 1 \subseteq \R$
From Closed Real Interval is Closed Set, $\closedint {\dfrac 1 n} 1$ is closed in $\R$ for all $n \in \N$.
But:
:$\ds \bigcup_{n \mathop \in \N} \closedint {\dfrac 1 n} 1 = \hointl 0 1$
The result follows from Half-Open Real Interval is neith... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $V_1, V_2, V_3, \ldots$ be an [[Definition:Infinite Set|infinite set]] of [[Definition:Closed Set (Metric Space)|closed sets]] of $M$.
Then it is not necessarily the case that $\ds \bigcup_{n \mathop \in \N} V_n$ is itself a [[Definition:Clo... | Consider the [[Definition:Closed Real Interval|closed real interval]]:
:$\closedint {\dfrac 1 n} 1 \subseteq \R$
From [[Closed Real Interval is Closed Set]], $\closedint {\dfrac 1 n} 1$ is [[Definition:Closed Set (Metric Space)|closed]] in $\R$ for all $n \in \N$.
But:
:$\ds \bigcup_{n \mathop \in \N} \closedint {\d... | Infinite Union of Closed Sets of Metric Space may not be Closed | https://proofwiki.org/wiki/Infinite_Union_of_Closed_Sets_of_Metric_Space_may_not_be_Closed | https://proofwiki.org/wiki/Infinite_Union_of_Closed_Sets_of_Metric_Space_may_not_be_Closed | [
"Closed Sets",
"Set Union"
] | [
"Definition:Metric Space",
"Definition:Infinite Set",
"Definition:Closed Set/Metric Space",
"Definition:Closed Set/Metric Space"
] | [
"Definition:Real Interval/Closed",
"Closed Real Interval is Closed Set",
"Definition:Closed Set/Metric Space",
"Half-Open Real Interval is neither Open nor Closed"
] |
proofwiki-10247 | Open Sets of Cartesian Product of Metric Spaces under Chebyshev Distance | Let $M_1 = \struct {A_1, d_1}, M_2 = \struct {A_2, d_2}, \ldots, M_n = \struct {A_n, d_n}$ be metric spaces.
Let $\ds \AA = \prod_{i \mathop = 1}^n A_i$ be the cartesian product of $A_1, A_2, \ldots, A_n$.
Let $d_\infty: \AA \times \AA \to \R$ be the Chebyshev distance on $\AA$:
:$\ds \map {d_\infty} {x, y} = \max_{i \... | A set $U$ is open {{iff}} it is the neighborhood of each of its points.
That is:
:$\forall a \in U: \exists \delta \in \R_{>0}: \map {B_\delta} a \subseteq U$
where $\map {B_\delta} a$ denotes the open $\delta$-ball of $a$.
Let $I = \set {1, 2, \ldots, n}$.
For all $i \in I$, let $U_i$ be open in $M_i$.
Then:
{{begin-e... | Let $M_1 = \struct {A_1, d_1}, M_2 = \struct {A_2, d_2}, \ldots, M_n = \struct {A_n, d_n}$ be [[Definition:Metric Space|metric spaces]].
Let $\ds \AA = \prod_{i \mathop = 1}^n A_i$ be the [[Definition:Finite Cartesian Product|cartesian product]] of $A_1, A_2, \ldots, A_n$.
Let $d_\infty: \AA \times \AA \to \R$ be the... | A set $U$ is [[Definition:Open Set (Metric Space)|open]] {{iff}} it is the [[Definition:Neighborhood (Metric Space)|neighborhood]] of each of its points.
That is:
:$\forall a \in U: \exists \delta \in \R_{>0}: \map {B_\delta} a \subseteq U$
where $\map {B_\delta} a$ denotes the [[Definition:Open Ball of Metric Space|o... | Open Sets of Cartesian Product of Metric Spaces under Chebyshev Distance | https://proofwiki.org/wiki/Open_Sets_of_Cartesian_Product_of_Metric_Spaces_under_Chebyshev_Distance | https://proofwiki.org/wiki/Open_Sets_of_Cartesian_Product_of_Metric_Spaces_under_Chebyshev_Distance | [
"Chebyshev Distance",
"Open Sets (Metric Spaces)"
] | [
"Definition:Metric Space",
"Definition:Cartesian Product/Finite",
"Definition:Chebyshev Distance",
"Definition:Open Set/Metric Space",
"Definition:Open Set/Metric Space"
] | [
"Definition:Open Set/Metric Space",
"Definition:Neighborhood (Metric Space)",
"Definition:Open Ball",
"Definition:Open Set/Metric Space",
"Definition:Open Set/Metric Space"
] |
proofwiki-10248 | Basis for Product of Metric Spaces under Chebyshev Distance | Let $M_1 = \struct {A_1, d_1}, M_2 = \struct {A_2, d_2}, \ldots, M_n = \struct {A_n, d_n}$ be metric spaces.
Let $\ds \AA = \prod_{i \mathop = 1}^n A_i$ be the cartesian product of $A_1, A_2, \ldots, A_n$.
Let $d_\infty: \AA \times \AA \to \R$ be the Chebyshev distance on $\AA$:
:$\ds \map {d_\infty} {x, y} = \max_{i \... | Let $U$ be an open set of $M$.
Then for all $a \in U$, we have:
:$\map {B_\delta} a \subseteq U$
for some $\delta \in \R_{>0}$.
Then:
:$\ds \bigcup_{i \mathop = 1}^n \map {B_\delta} {a_i} \subseteq U$
and the result follows.
{{qed}} | Let $M_1 = \struct {A_1, d_1}, M_2 = \struct {A_2, d_2}, \ldots, M_n = \struct {A_n, d_n}$ be [[Definition:Metric Space|metric spaces]].
Let $\ds \AA = \prod_{i \mathop = 1}^n A_i$ be the [[Definition:Finite Cartesian Product|cartesian product]] of $A_1, A_2, \ldots, A_n$.
Let $d_\infty: \AA \times \AA \to \R$ be the... | Let $U$ be an [[Definition:Open Set (Metric Space)|open set]] of $M$.
Then for all $a \in U$, we have:
:$\map {B_\delta} a \subseteq U$
for some $\delta \in \R_{>0}$.
Then:
:$\ds \bigcup_{i \mathop = 1}^n \map {B_\delta} {a_i} \subseteq U$
and the result follows.
{{qed}} | Basis for Product of Metric Spaces under Chebyshev Distance | https://proofwiki.org/wiki/Basis_for_Product_of_Metric_Spaces_under_Chebyshev_Distance | https://proofwiki.org/wiki/Basis_for_Product_of_Metric_Spaces_under_Chebyshev_Distance | [
"Chebyshev Distance",
"Open Sets (Metric Spaces)"
] | [
"Definition:Metric Space",
"Definition:Cartesian Product/Finite",
"Definition:Chebyshev Distance",
"Definition:Open Set/Metric Space",
"Definition:Basis for Open Sets (Metric Space)"
] | [
"Definition:Open Set/Metric Space"
] |
proofwiki-10249 | Open Balls form Basis for Open Sets of Metric Space | Let $M = \struct {A, d}$ be a metric space.
Let $\BB$ be the set of all open balls of $M$.
Then $\BB$ is a basis for the open sets of $M$. | Let $U$ be an open set of $M$.
Then by definition:
:$\forall y \in U: \exists \epsilon \in \R_{>0}: \map {B_\epsilon} y \subseteq U$
Thus:
:$\ds U = \bigcup_{y \mathop \in U} \map {B_\epsilon} y$
{{qed}} | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $\BB$ be the [[Definition:Set|set]] of all [[Definition:Open Ball|open balls]] of $M$.
Then $\BB$ is a [[Definition:Basis for Open Sets (Metric Space)|basis for the open sets]] of $M$. | Let $U$ be an [[Definition:Open Set (Metric Space)|open set]] of $M$.
Then by definition:
:$\forall y \in U: \exists \epsilon \in \R_{>0}: \map {B_\epsilon} y \subseteq U$
Thus:
:$\ds U = \bigcup_{y \mathop \in U} \map {B_\epsilon} y$
{{qed}} | Open Balls form Basis for Open Sets of Metric Space | https://proofwiki.org/wiki/Open_Balls_form_Basis_for_Open_Sets_of_Metric_Space | https://proofwiki.org/wiki/Open_Balls_form_Basis_for_Open_Sets_of_Metric_Space | [
"Open Sets (Metric Spaces)",
"Open Balls",
"Topological Bases"
] | [
"Definition:Metric Space",
"Definition:Set",
"Definition:Open Ball",
"Definition:Basis for Open Sets (Metric Space)"
] | [
"Definition:Open Set/Metric Space"
] |
proofwiki-10250 | Graph of Continuous Mapping between Metric Spaces is Closed in Chebyshev Product | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.
Let $f: A_1 \to A_2$ be a continuous mapping.
Let $\AA = A_1 \times A_2$ be the cartesian product of $A_1$ and $A_2$.
Let $d_\infty: \AA \times \AA \to \R$ be the Chebyshev distance on $\AA$:
:$\ds \map {d_\infty} {x, y} = \max \set {\map {... | Let $f: A_1 \to A_2$ be continuous.
Then:
:$\forall a \in A_1: \ds \lim_{n \mathop \to \infty} x_n = a \implies \lim_{n \mathop \to \infty} \map f {x_n} = \map f a$
This can be extended to ordered pairs and ordered tuples, because:
:$\ds \lim_{n \mathop \to \infty} x_n = a \iff \exists N \in \N: n > N \implies x_n \in ... | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be [[Definition:Metric Space|metric spaces]].
Let $f: A_1 \to A_2$ be a [[Definition:Continuous Mapping (Metric Spaces)|continuous mapping]].
Let $\AA = A_1 \times A_2$ be the [[Definition:Cartesian Product|cartesian product]] of $A_1$ and $A_2$.
Let $d_\... | Let $f: A_1 \to A_2$ be [[Definition:Continuous Mapping (Metric Spaces)|continuous]].
Then:
:$\forall a \in A_1: \ds \lim_{n \mathop \to \infty} x_n = a \implies \lim_{n \mathop \to \infty} \map f {x_n} = \map f a$
This can be extended to [[Definition:Ordered Pair|ordered pairs]] and [[Definition:Ordered Tuple|ordere... | Graph of Continuous Mapping between Metric Spaces is Closed in Chebyshev Product | https://proofwiki.org/wiki/Graph_of_Continuous_Mapping_between_Metric_Spaces_is_Closed_in_Chebyshev_Product | https://proofwiki.org/wiki/Graph_of_Continuous_Mapping_between_Metric_Spaces_is_Closed_in_Chebyshev_Product | [
"Chebyshev Distance",
"Continuous Mappings"
] | [
"Definition:Metric Space",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Cartesian Product",
"Definition:Chebyshev Distance",
"Definition:Graph of Mapping",
"Definition:Closed Set/Metric Space"
] | [
"Definition:Continuous Mapping (Metric Space)",
"Definition:Ordered Pair",
"Definition:Ordered Tuple",
"Definition:Neighborhood (Metric Space)",
"Definition:Sequence",
"Definition:Convergent Sequence/Metric Space",
"Definition:Limit Point/Metric Space",
"Definition:Closed Set/Metric Space"
] |
proofwiki-10251 | Closed Subset of Real Numbers with Lower Bound contains Infimum | Consider the real number line as a metric space under the usual metric.
Let $A \subseteq \R$ such that $A$ is closed in $\R$ and $A \ne \O$.
Let $A$ be bounded below.
Then $A$ contains its infimum. | From Infimum of Bounded Below Set of Reals is in Closure:
:$\inf A \in \map \cl A$
From Set is Closed iff Equals Topological Closure:
:$A = \map \cl A$
Therefore $\inf A \in A$.
{{qed}} | Consider the [[Definition:Real Number Line|real number line]] as a [[Definition:Metric Space|metric space]] under the [[Definition:Usual Metric|usual metric]].
Let $A \subseteq \R$ such that $A$ is [[Definition:Closed Set (Metric Space)|closed in $\R$]] and $A \ne \O$.
Let $A$ be [[Definition:Bounded Below Subset of ... | From [[Infimum of Bounded Below Set of Reals is in Closure]]:
:$\inf A \in \map \cl A$
From [[Set is Closed iff Equals Topological Closure]]:
:$A = \map \cl A$
Therefore $\inf A \in A$.
{{qed}} | Closed Subset of Real Numbers with Lower Bound contains Infimum | https://proofwiki.org/wiki/Closed_Subset_of_Real_Numbers_with_Lower_Bound_contains_Infimum | https://proofwiki.org/wiki/Closed_Subset_of_Real_Numbers_with_Lower_Bound_contains_Infimum | [
"Bounded Below Sets of Real Numbers",
"Real Analysis"
] | [
"Definition:Real Number/Real Number Line",
"Definition:Metric Space",
"Definition:Usual Metric",
"Definition:Closed Set/Metric Space",
"Definition:Bounded Below Set/Real Numbers",
"Definition:Infimum of Set/Real Numbers"
] | [
"Infimum of Bounded Below Set of Reals is in Closure",
"Set is Closed iff Equals Topological Closure"
] |
proofwiki-10252 | Set of Isolated Points of Metric Space is Disjoint from Limit Points | Let $M = \struct {A, d}$ be a metric space.
Let $H \subseteq A$ be a subset of $A$.
Let $H'$ be the set of limit points of $H$.
Let $H^i$ be the set of isolated points of $H$.
Then:
:$H' \cap H^i = \O$ | Let $a \in H_i$.
Then by definition of isolated point:
:$\exists \epsilon \in \R_{>0}: \set {x \in H: \map d {x, a} < \epsilon} = \set a$
But by {{Metric-space-axiom|1}}:
:$\map d {a, a} = 0$
and so:
:$\set {x \in H: 0 < \map d {x, a} < \epsilon} = \O$
So by definition $a$ is not a limit point of $H$.
That is:
:$a \not... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $H \subseteq A$ be a [[Definition:Subset|subset]] of $A$.
Let $H'$ be the [[Definition:Set|set]] of [[Definition:Limit Point (Metric Space)|limit points]] of $H$.
Let $H^i$ be the [[Definition:Set|set]] of [[Definition:Isolated Point of Subs... | Let $a \in H_i$.
Then by definition of [[Definition:Isolated Point of Subset of Metric Space|isolated point]]:
:$\exists \epsilon \in \R_{>0}: \set {x \in H: \map d {x, a} < \epsilon} = \set a$
But by {{Metric-space-axiom|1}}:
:$\map d {a, a} = 0$
and so:
:$\set {x \in H: 0 < \map d {x, a} < \epsilon} = \O$
So by d... | Set of Isolated Points of Metric Space is Disjoint from Limit Points | https://proofwiki.org/wiki/Set_of_Isolated_Points_of_Metric_Space_is_Disjoint_from_Limit_Points | https://proofwiki.org/wiki/Set_of_Isolated_Points_of_Metric_Space_is_Disjoint_from_Limit_Points | [
"Limit Points",
"Isolated Points"
] | [
"Definition:Metric Space",
"Definition:Subset",
"Definition:Set",
"Definition:Limit Point/Metric Space",
"Definition:Set",
"Definition:Isolated Point (Metric Space)/Subset"
] | [
"Definition:Isolated Point (Metric Space)/Subset",
"Definition:Limit Point/Metric Space",
"Intersection with Complement is Empty iff Subset"
] |
proofwiki-10253 | Subset of Metric Space is Subset of its Closure | Let $M = \struct {A, d}$ be a metric space.
Let $H \subseteq A$ be a subset of $A$.
Then:
:$H \subseteq H^-$
where $H^-$ denotes the closure of $H$. | By definition of closure:
:$H^- = H' \cup H^i$
where:
:$H'$ denotes the set of limit points of $H$
:$H^i$ denotes the set of isolated points of $H$.
Let $a \in H$.
If $a$ is a limit point of $H$ then $a \in H'$.
Suppose $a \notin H'$.
Then by definition of limit point:
:$\neg \forall \epsilon \in \R_{>0}: \set {x \in A... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $H \subseteq A$ be a [[Definition:Subset|subset]] of $A$.
Then:
:$H \subseteq H^-$
where $H^-$ denotes the [[Definition:Closure (Metric Space)|closure]] of $H$. | By definition of [[Definition:Closure (Metric Space)|closure]]:
:$H^- = H' \cup H^i$
where:
:$H'$ denotes the [[Definition:Set|set]] of [[Definition:Limit Point (Metric Space)|limit points]] of $H$
:$H^i$ denotes the [[Definition:Set|set]] of [[Definition:Isolated Point of Subset of Metric Space|isolated points]] of $... | Subset of Metric Space is Subset of its Closure | https://proofwiki.org/wiki/Subset_of_Metric_Space_is_Subset_of_its_Closure | https://proofwiki.org/wiki/Subset_of_Metric_Space_is_Subset_of_its_Closure | [
"Set Closures"
] | [
"Definition:Metric Space",
"Definition:Subset",
"Definition:Closure (Topology)/Metric Space"
] | [
"Definition:Closure (Topology)/Metric Space",
"Definition:Set",
"Definition:Limit Point/Metric Space",
"Definition:Set",
"Definition:Isolated Point (Metric Space)/Subset",
"Definition:Limit Point/Metric Space",
"Definition:Limit Point/Metric Space",
"Definition:Isolated Point (Metric Space)/Subset",
... |
proofwiki-10254 | Point in Closure of Subset of Metric Space iff Limit of Sequence | Let $M = \struct {A, d}$ be a metric space.
Let $H \subseteq A$ be a subset of $A$.
Let $H^-$ denote the closure of $H$.
Let $a \in A$.
Then $a \in H^-$ {{iff}} there exists a sequence $\sequence {x_n}$ of points of $H$ which converges to the limit $a$. | From definition of closure, $H^- = H' \cup H^i$.
Suppose that $a \in H^-$.
If $a \in H^i$, then $a \in H$ and so $\sequence {a, a, \ldots}$ is a sequence in $H$ that converges to $a$.
If $a \in H'$, then by {{Defof|Limit Point (Metric Space)}} there exists a sequence in $H$ that converges to $a$.
{{qed|lemma}}
For the ... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $H \subseteq A$ be a [[Definition:Subset|subset]] of $A$.
Let $H^-$ denote the [[Definition:Closure (Metric Space)|closure]] of $H$.
Let $a \in A$.
Then $a \in H^-$ {{iff}} there exists a [[Definition:Sequence|sequence]] $\sequence {x_n}$ ... | From definition of [[Definition:Closure (Metric Space)|closure]], $H^- = H' \cup H^i$.
Suppose that $a \in H^-$.
If $a \in H^i$, then $a \in H$ and so $\sequence {a, a, \ldots}$ is a [[Definition:Sequence|sequence]] in $H$ that converges to $a$.
If $a \in H'$, then by {{Defof|Limit Point (Metric Space)}} there exis... | Point in Closure of Subset of Metric Space iff Limit of Sequence | https://proofwiki.org/wiki/Point_in_Closure_of_Subset_of_Metric_Space_iff_Limit_of_Sequence | https://proofwiki.org/wiki/Point_in_Closure_of_Subset_of_Metric_Space_iff_Limit_of_Sequence | [
"Set Closures"
] | [
"Definition:Metric Space",
"Definition:Subset",
"Definition:Closure (Topology)/Metric Space",
"Definition:Sequence",
"Definition:Convergent Sequence/Metric Space",
"Definition:Limit of Sequence/Metric Space"
] | [
"Definition:Closure (Topology)/Metric Space",
"Definition:Sequence",
"Definition:Sequence",
"Definition:Convergent Sequence/Metric Space",
"Definition:Sequence",
"Definition:Convergent Sequence/Metric Space",
"Definition:Open Set/Metric Space",
"Definition:Open Set/Metric Space",
"Definition:Open Ba... |
proofwiki-10255 | Closure of Subset of Closed Set of Metric Space is Subset | Let $M = \struct {A, d}$ be a metric space.
Let $F$ be a closed set of $M$.
Let $H \subseteq F$ be a subset of $F$.
Let $H^-$ denote the closure of $H$.
Then $H^- \subseteq F$. | From Metric Induces Topology, the topology $\tau$ induced by the metric $d$ is a topology on $M$.
From Metric Closure and Topological Closure of Subset are Equivalent, it is sufficient to show that the topological closure of $H$ is contained in $F$.
From Set is Closed in Metric Space iff Closed in Induced Topological S... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $F$ be a [[Definition:Closed Set (Metric Space)|closed set]] of $M$.
Let $H \subseteq F$ be a [[Definition:Subset|subset]] of $F$.
Let $H^-$ denote the [[Definition:Closure (Metric Space)|closure]] of $H$.
Then $H^- \subseteq F$. | From [[Metric Induces Topology]], the [[Definition:Topology Induced by Metric|topology $\tau$ induced]] by the [[Definition:Metric|metric]] $d$ is a [[Definition:Topology|topology]] on $M$.
From [[Metric Closure and Topological Closure of Subset are Equivalent]], it is sufficient to show that the [[Definition:Topologi... | Closure of Subset of Closed Set of Metric Space is Subset/Proof 1 | https://proofwiki.org/wiki/Closure_of_Subset_of_Closed_Set_of_Metric_Space_is_Subset | https://proofwiki.org/wiki/Closure_of_Subset_of_Closed_Set_of_Metric_Space_is_Subset/Proof_1 | [
"Set Closures",
"Closure of Subset of Closed Set of Metric Space is Subset"
] | [
"Definition:Metric Space",
"Definition:Closed Set/Metric Space",
"Definition:Subset",
"Definition:Closure (Topology)/Metric Space"
] | [
"Metric Induces Topology",
"Definition:Topology Induced by Metric",
"Definition:Metric Space/Metric",
"Definition:Topology",
"Metric Closure and Topological Closure of Subset are Equivalent",
"Definition:Closure (Topology)",
"Set is Closed in Metric Space iff Closed in Induced Topological Space",
"Def... |
proofwiki-10256 | Closure of Subset of Closed Set of Metric Space is Subset | Let $M = \struct {A, d}$ be a metric space.
Let $F$ be a closed set of $M$.
Let $H \subseteq F$ be a subset of $F$.
Let $H^-$ denote the closure of $H$.
Then $H^- \subseteq F$. | Let $x \in H^-$.
From Point in Closure of Subset of Metric Space iff Limit of Sequence
:there exists a sequence $\sequence {a_n}$ of points of $H$ which converges to the limit $x$.
By assumption:
:$\sequence {a_n}$ is also a sequence of points of $F$
From Subset of Metric Space contains Limits of Sequences iff Closed:
... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $F$ be a [[Definition:Closed Set (Metric Space)|closed set]] of $M$.
Let $H \subseteq F$ be a [[Definition:Subset|subset]] of $F$.
Let $H^-$ denote the [[Definition:Closure (Metric Space)|closure]] of $H$.
Then $H^- \subseteq F$. | Let $x \in H^-$.
From [[Point in Closure of Subset of Metric Space iff Limit of Sequence]]
:there exists a [[Definition:Sequence|sequence]] $\sequence {a_n}$ of points of $H$ which [[Definition:Convergent Sequence in Metric Space|converges]] to the [[Definition:Limit of Sequence (Metric Space)|limit]] $x$.
By assumpt... | Closure of Subset of Closed Set of Metric Space is Subset/Proof 2 | https://proofwiki.org/wiki/Closure_of_Subset_of_Closed_Set_of_Metric_Space_is_Subset | https://proofwiki.org/wiki/Closure_of_Subset_of_Closed_Set_of_Metric_Space_is_Subset/Proof_2 | [
"Set Closures",
"Closure of Subset of Closed Set of Metric Space is Subset"
] | [
"Definition:Metric Space",
"Definition:Closed Set/Metric Space",
"Definition:Subset",
"Definition:Closure (Topology)/Metric Space"
] | [
"Point in Closure of Subset of Metric Space iff Limit of Sequence",
"Definition:Sequence",
"Definition:Convergent Sequence/Metric Space",
"Definition:Limit of Sequence/Metric Space",
"Definition:Sequence",
"Subset of Metric Space contains Limits of Sequences iff Closed"
] |
proofwiki-10257 | Closure of Subset of Metric Space is Intersection of Closed Supersets | Let $M = \struct {A, d}$ be a metric space.
Let $H \subseteq A$ be a subset of $A$.
Let $H^-$ denote the closure of $H$.
Then $H^-$ is the intersection of all closed sets of $M$ of which $H$ is a subset. | Let $\mathbb K$ be the set of all closed sets $K$ of $M$ such that $H \subseteq K$.
From Closure of Subset of Closed Set of Metric Space is Subset:
:$\forall K \in \mathbb K: H^- \subseteq K$
From Intersection is Largest Subset:
:$\ds H^- \subseteq \bigcap \mathbb K$
{{qed|lemma}}
From Closure of Subset of Metric Space... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $H \subseteq A$ be a [[Definition:Subset|subset]] of $A$.
Let $H^-$ denote the [[Definition:Closure (Metric Space)|closure]] of $H$.
Then $H^-$ is the [[Definition:Set Intersection|intersection]] of all [[Definition:Closed Set (Metric Space... | Let $\mathbb K$ be the [[Definition:Set|set]] of all [[Definition:Closed Set (Metric Space)|closed sets]] $K$ of $M$ such that $H \subseteq K$.
From [[Closure of Subset of Closed Set of Metric Space is Subset]]:
:$\forall K \in \mathbb K: H^- \subseteq K$
From [[Intersection is Largest Subset]]:
:$\ds H^- \subseteq ... | Closure of Subset of Metric Space is Intersection of Closed Supersets | https://proofwiki.org/wiki/Closure_of_Subset_of_Metric_Space_is_Intersection_of_Closed_Supersets | https://proofwiki.org/wiki/Closure_of_Subset_of_Metric_Space_is_Intersection_of_Closed_Supersets | [
"Set Closures"
] | [
"Definition:Metric Space",
"Definition:Subset",
"Definition:Closure (Topology)/Metric Space",
"Definition:Set Intersection",
"Definition:Closed Set/Metric Space",
"Definition:Subset"
] | [
"Definition:Set",
"Definition:Closed Set/Metric Space",
"Closure of Subset of Closed Set of Metric Space is Subset",
"Intersection is Largest Subset",
"Closure of Subset of Metric Space is Closed",
"Definition:Closed Set/Metric Space",
"Subset of Metric Space is Subset of its Closure",
"Definition:Clo... |
proofwiki-10258 | Rational Numbers form Metric Subspace of Real Numbers under Euclidean Metric | Let $\struct {\Q, d_\Q}$ be the set of rational numbers under the function $d_\Q: \Q \times \Q \to \R$ defined as:
:$\forall x, y \in \Q: \map {d_\Q} {x, y} = \size {x - y}$
Let $\struct {\R, d}$ denote the real number line with the usual (Euclidean) metric.
Then $\struct {\Q, d_\Q}$ is a metric subspace of $\struct {\... | From Rational Numbers form Subfield of Real Numbers:
:$\Q \subseteq \R$
By definition of the Euclidean metric on $\R$:
:$\forall x, y \in \R: \map d {x, y} = \size {x - y}$
and so $d_\Q$ is a restriction of $d$:
:$d_\Q = d {\restriction}_{\Q \times \Q}$
From Euclidean Metric on Real Number Line is Metric, $d$ is a metr... | Let $\struct {\Q, d_\Q}$ be the [[Definition:Rational Number|set of rational numbers]] under the [[Definition:Real-Valued Function|function]] $d_\Q: \Q \times \Q \to \R$ defined as:
:$\forall x, y \in \Q: \map {d_\Q} {x, y} = \size {x - y}$
Let $\struct {\R, d}$ denote the [[Definition:Real Number Line with Euclidean ... | From [[Rational Numbers form Subfield of Real Numbers]]:
:$\Q \subseteq \R$
By definition of the [[Definition:Euclidean Metric on Real Number Line|Euclidean metric]] on $\R$:
:$\forall x, y \in \R: \map d {x, y} = \size {x - y}$
and so $d_\Q$ is a [[Definition:Restriction of Mapping|restriction]] of $d$:
:$d_\Q = d {... | Rational Numbers form Metric Subspace of Real Numbers under Euclidean Metric | https://proofwiki.org/wiki/Rational_Numbers_form_Metric_Subspace_of_Real_Numbers_under_Euclidean_Metric | https://proofwiki.org/wiki/Rational_Numbers_form_Metric_Subspace_of_Real_Numbers_under_Euclidean_Metric | [
"Rational Number Space",
"Real Number Line with Euclidean Metric"
] | [
"Definition:Rational Number",
"Definition:Real-Valued Function",
"Definition:Euclidean Metric/Real Number Line",
"Definition:Metric Subspace"
] | [
"Rational Numbers form Subfield of Real Numbers",
"Definition:Euclidean Metric/Real Number Line",
"Definition:Restriction/Mapping",
"Euclidean Metric on Real Number Line is Metric",
"Definition:Metric Space/Metric",
"Definition:Metric Subspace"
] |
proofwiki-10259 | Euclidean Metric on Real Number Line is Metric | The Euclidean metric on the real number line $\R$ is a metric. | The Euclidean metric on the real number line is a special case of the Euclidean metric on the real vector space $\R^n$.
The result follows from Euclidean Metric on Real Vector Space is Metric.
{{qed}} | The [[Definition:Euclidean Metric on Real Number Line|Euclidean metric]] on the [[Definition:Real Number Line|real number line]] $\R$ is a [[Definition:Metric|metric]]. | The [[Definition:Euclidean Metric on Real Number Line|Euclidean metric]] on the [[Definition:Real Number Line|real number line]] is a special case of the [[Definition:Euclidean Metric on Real Vector Space|Euclidean metric]] on the [[Definition:Real Vector Space|real vector space]] $\R^n$.
The result follows from [[Euc... | Euclidean Metric on Real Number Line is Metric/Proof 1 | https://proofwiki.org/wiki/Euclidean_Metric_on_Real_Number_Line_is_Metric | https://proofwiki.org/wiki/Euclidean_Metric_on_Real_Number_Line_is_Metric/Proof_1 | [
"Euclidean Metric",
"Real Number Line with Euclidean Metric",
"Euclidean Metric on Real Number Line is Metric"
] | [
"Definition:Euclidean Metric/Real Number Line",
"Definition:Real Number/Real Number Line",
"Definition:Metric Space/Metric"
] | [
"Definition:Euclidean Metric/Real Number Line",
"Definition:Real Number/Real Number Line",
"Definition:Euclidean Metric/Real Vector Space",
"Definition:Real Vector Space",
"Euclidean Metric on Real Vector Space is Metric"
] |
proofwiki-10260 | Euclidean Metric on Real Number Line is Metric | The Euclidean metric on the real number line $\R$ is a metric. | Consider the real number line under the Euclidean metric:
:$M = \struct {\R, d}$
where $d$ is the distance function given by:
:$\map d {x, y} = \size {x - y}$
=== Proof of {{Metric-space-axiom|1|nolink}} ===
{{begin-eqn}}
{{eqn | l = \map d {x, x}
| r = \size {x - x}
| c = Definition of $d$
}}
{{eqn | r = \... | The [[Definition:Euclidean Metric on Real Number Line|Euclidean metric]] on the [[Definition:Real Number Line|real number line]] $\R$ is a [[Definition:Metric|metric]]. | Consider the [[Definition:Real Number Line|real number line]] under the [[Definition:Euclidean Metric on Real Number Line|Euclidean metric]]:
:$M = \struct {\R, d}$
where $d$ is the [[Definition:Distance Function|distance function]] given by:
:$\map d {x, y} = \size {x - y}$
=== Proof of {{Metric-space-axiom|1|nolin... | Euclidean Metric on Real Number Line is Metric/Proof 2 | https://proofwiki.org/wiki/Euclidean_Metric_on_Real_Number_Line_is_Metric | https://proofwiki.org/wiki/Euclidean_Metric_on_Real_Number_Line_is_Metric/Proof_2 | [
"Euclidean Metric",
"Real Number Line with Euclidean Metric",
"Euclidean Metric on Real Number Line is Metric"
] | [
"Definition:Euclidean Metric/Real Number Line",
"Definition:Real Number/Real Number Line",
"Definition:Metric Space/Metric"
] | [
"Definition:Real Number/Real Number Line",
"Definition:Euclidean Metric/Real Number Line",
"Definition:Distance Function",
"Triangle Inequality/Real Numbers"
] |
proofwiki-10261 | Unit n-Cube under Chebyshev Distance is Subspace of Real Vector Space | Let $n \in \N$.
Let $I^n$ denote the unit $n$-cube:
:$I^n = \closedint 0 1^n$
that is, the Cartesian product of $n$ instances of the closed real interval $\set {x \in \R: 0 \le x \le 1}$.
Let $d_c: I^n \times I^n \to \R$ be defined as:
:$\ds \map {d_c} {x, y} = \max_{i \mathop = 1}^n \set {\size {x_i - y_i} }$
where $x... | {{ProofWanted|Straightforward but a bit tedious.}} | Let $n \in \N$.
Let $I^n$ denote the [[Definition:Unit n-Cube|unit $n$-cube]]:
:$I^n = \closedint 0 1^n$
that is, the [[Definition:Finite Cartesian Product|Cartesian product]] of $n$ instances of the [[Definition:Closed Real Interval|closed real interval]] $\set {x \in \R: 0 \le x \le 1}$.
Let $d_c: I^n \times I^n \... | {{ProofWanted|Straightforward but a bit tedious.}} | Unit n-Cube under Chebyshev Distance is Subspace of Real Vector Space | https://proofwiki.org/wiki/Unit_n-Cube_under_Chebyshev_Distance_is_Subspace_of_Real_Vector_Space | https://proofwiki.org/wiki/Unit_n-Cube_under_Chebyshev_Distance_is_Subspace_of_Real_Vector_Space | [
"Metric Subspaces"
] | [
"Definition:Unit n-Cube",
"Definition:Cartesian Product/Finite",
"Definition:Real Interval/Closed",
"Definition:Metric Subspace",
"Definition:Chebyshev Distance/Real Vector Space",
"Definition:Real Vector Space"
] | [] |
proofwiki-10262 | Unit n-Sphere under Euclidean Metric is Metric Subspace of Euclidean Real Vector Space | Let $\Bbb S^n$ be the unit $n$-sphere.
Let $d_S: \Bbb S^n \times \Bbb S^n \to \R$ be the real-valued function defined as:
:$\ds \forall x, y \in \Bbb S^n: \map {d_S} {x, y} = \sqrt {\sum_{i \mathop = 1}^{n + 1} \paren {x_i - y_i}^2}$
where $x = \tuple {x_1, x_2, \ldots, x_{n + 1} }, y = \tuple {y_1, y_2, \ldots, y_{n +... | The metric given is the Euclidean metric restricted to the subset $\Bbb S^n$ of the real vector space $\R^{n + 1}$.
The result follows from Subspace of Metric Space is Metric Space.
{{qed}} | Let $\Bbb S^n$ be the [[Definition:Unit Sphere (Topology)|unit $n$-sphere]].
Let $d_S: \Bbb S^n \times \Bbb S^n \to \R$ be the [[Definition:Real-Valued Function|real-valued function]] defined as:
:$\ds \forall x, y \in \Bbb S^n: \map {d_S} {x, y} = \sqrt {\sum_{i \mathop = 1}^{n + 1} \paren {x_i - y_i}^2}$
where $x = ... | The [[Definition:Metric|metric]] given is the [[Definition:Euclidean Metric on Real Vector Space|Euclidean metric]] [[Definition:Restriction of Mapping|restricted]] to the [[Definition:Subset|subset]] $\Bbb S^n$ of the [[Definition:Real Vector Space|real vector space]] $\R^{n + 1}$.
The result follows from [[Subspace ... | Unit n-Sphere under Euclidean Metric is Metric Subspace of Euclidean Real Vector Space | https://proofwiki.org/wiki/Unit_n-Sphere_under_Euclidean_Metric_is_Metric_Subspace_of_Euclidean_Real_Vector_Space | https://proofwiki.org/wiki/Unit_n-Sphere_under_Euclidean_Metric_is_Metric_Subspace_of_Euclidean_Real_Vector_Space | [
"Metric Subspaces"
] | [
"Definition:Unit Sphere/Topology",
"Definition:Real-Valued Function",
"Definition:Metric Subspace",
"Definition:Euclidean Metric/Real Vector Space",
"Definition:Real Vector Space"
] | [
"Definition:Metric Space/Metric",
"Definition:Euclidean Metric/Real Vector Space",
"Definition:Restriction/Mapping",
"Definition:Subset",
"Definition:Real Vector Space",
"Subspace of Metric Space is Metric Space"
] |
proofwiki-10263 | Vector Subspace of Real Vector Space under Chebyshev Metric is Metric Subspace | Let $n \in \N$.
Let $A$ be the set of all ordered $n+1$-tuples $\tuple {x_1, x_2, \ldots, x_{n + 1} }$ of real numbers such that $x_{n + 1} = 0$.
Let $d: A \times A \to \R$ be the function defined as:
:$\ds \forall x, y \in A: \map d {x, y} = \max_{i \mathop = 1}^n \set {\size {x_i - y_i} }$
where $x = \tuple {x_1, x_2... | The metric given is the Chebyshev distance restricted to the subset $A$ of the real vector space $\R^{n + 1}$.
The result follows from Subspace of Metric Space is Metric Space.
{{qed}} | Let $n \in \N$.
Let $A$ be the set of all [[Definition:Ordered Tuple|ordered $n+1$-tuples]] $\tuple {x_1, x_2, \ldots, x_{n + 1} }$ of [[Definition:Real Number|real numbers]] such that $x_{n + 1} = 0$.
Let $d: A \times A \to \R$ be the [[Definition:Real-Valued Function|function]] defined as:
:$\ds \forall x, y \in A:... | The [[Definition:Metric|metric]] given is the [[Definition:Chebyshev Distance on Real Vector Space|Chebyshev distance]] [[Definition:Restriction of Mapping|restricted]] to the [[Definition:Subset|subset]] $A$ of the [[Definition:Real Vector Space|real vector space]] $\R^{n + 1}$.
The result follows from [[Subspace of ... | Vector Subspace of Real Vector Space under Chebyshev Metric is Metric Subspace | https://proofwiki.org/wiki/Vector_Subspace_of_Real_Vector_Space_under_Chebyshev_Metric_is_Metric_Subspace | https://proofwiki.org/wiki/Vector_Subspace_of_Real_Vector_Space_under_Chebyshev_Metric_is_Metric_Subspace | [
"Metric Subspaces",
"Chebyshev Distance"
] | [
"Definition:Ordered Tuple",
"Definition:Real Number",
"Definition:Real-Valued Function",
"Definition:Metric Subspace",
"Definition:Chebyshev Distance/Real Vector Space",
"Definition:Real Vector Space"
] | [
"Definition:Metric Space/Metric",
"Definition:Chebyshev Distance/Real Vector Space",
"Definition:Restriction/Mapping",
"Definition:Subset",
"Definition:Real Vector Space",
"Subspace of Metric Space is Metric Space"
] |
proofwiki-10264 | Inclusion Mapping on Metric Space is Continuous | Let $M = \struct {A, d}$ be a metric space.
Let $\struct {H, d_H}$ be a metric subspace of $M$.
Then the inclusion mapping $i_H: H \to A$ is continuous from $\struct {H, d_H}$ to $\struct {A, d}$. | Let $a \in H$.
Let $\epsilon \in \R_{>0}$.
Let $\delta = \epsilon$.
Then:
{{begin-eqn}}
{{eqn | l = \map {d_H} {a, y}
| o = <
| r = \delta
| c = for some $y \in H$
}}
{{eqn | ll= \leadsto
| l = \map d {\map {i_H} a, \map {i_H} y}
| r = \map d {a, y}
| c = {{Defof|Inclusion Mapping}}
... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $\struct {H, d_H}$ be a [[Definition:Metric Subspace|metric subspace]] of $M$.
Then the [[Definition:Inclusion Mapping|inclusion mapping]] $i_H: H \to A$ is [[Definition:Continuous on Metric Space|continuous]] from $\struct {H, d_H}$ to $\st... | Let $a \in H$.
Let $\epsilon \in \R_{>0}$.
Let $\delta = \epsilon$.
Then:
{{begin-eqn}}
{{eqn | l = \map {d_H} {a, y}
| o = <
| r = \delta
| c = for some $y \in H$
}}
{{eqn | ll= \leadsto
| l = \map d {\map {i_H} a, \map {i_H} y}
| r = \map d {a, y}
| c = {{Defof|Inclusion Mappi... | Inclusion Mapping on Metric Space is Continuous | https://proofwiki.org/wiki/Inclusion_Mapping_on_Metric_Space_is_Continuous | https://proofwiki.org/wiki/Inclusion_Mapping_on_Metric_Space_is_Continuous | [
"Inclusion Mappings",
"Metric Subspaces",
"Continuous Mappings"
] | [
"Definition:Metric Space",
"Definition:Metric Subspace",
"Definition:Inclusion Mapping",
"Definition:Continuous Mapping (Metric Space)/Space"
] | [
"Definition:Continuous Mapping (Metric Space)/Point",
"Definition:Continuous Mapping (Metric Space)/Space"
] |
proofwiki-10265 | Bertrand-Chebyshev Theorem | For all $n \in \N_{>0}$, there exists a prime number $p$ with $n < p \le 2 n$. | We will first prove the theorem for the case $n \le 2047$.
Consider the following sequence of prime numbers:
:$2, 3, 5, 7, 13, 23, 43, 83, 163, 317, 631, 1259, 2503$
Each of these prime number is smaller than twice the previous one.
Hence every interval $\set {x: n < x \le 2 n}$, with $n \le 2047$, contains one of the... | For all $n \in \N_{>0}$, there exists a [[Definition:Prime Number|prime number]] $p$ with $n < p \le 2 n$. | We will first prove the theorem for the case $n \le 2047$.
Consider the following sequence of [[Definition:Prime Number|prime numbers]]:
:$2, 3, 5, 7, 13, 23, 43, 83, 163, 317, 631, 1259, 2503$
Each of these [[Definition:Prime Number|prime number]] is smaller than twice the previous one.
Hence every interval $\set... | Bertrand-Chebyshev Theorem/Proof 1 | https://proofwiki.org/wiki/Bertrand-Chebyshev_Theorem | https://proofwiki.org/wiki/Bertrand-Chebyshev_Theorem/Proof_1 | [
"Bertrand-Chebyshev Theorem",
"Number Theory"
] | [
"Definition:Prime Number"
] | [
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Factor",
"Definition:Prime Number",
"Definition:Contradiction",
"Definition:Sufficiently Large"
] |
proofwiki-10266 | Bertrand-Chebyshev Theorem | For all $n \in \N_{>0}$, there exists a prime number $p$ with $n < p \le 2 n$. | We will first prove the theorem for the case $n \le 426$.
Consider the following sequence of prime numbers:
:$2, 3, 5, 7, 13, 23, 43, 83, 163, 317, 631$
Each of these prime number is smaller than twice the previous one.
Hence every interval $\set {x: n < x \le 2 n}$, with $n \le 426$, contains one of these prime numbe... | For all $n \in \N_{>0}$, there exists a [[Definition:Prime Number|prime number]] $p$ with $n < p \le 2 n$. | We will first prove the theorem for the case $n \le 426$.
Consider the following sequence of [[Definition:Prime Number|prime numbers]]:
:$2, 3, 5, 7, 13, 23, 43, 83, 163, 317, 631$
Each of these [[Definition:Prime Number|prime number]] is smaller than twice the previous one.
Hence every interval $\set {x: n < x \l... | Bertrand-Chebyshev Theorem/Proof 2 | https://proofwiki.org/wiki/Bertrand-Chebyshev_Theorem | https://proofwiki.org/wiki/Bertrand-Chebyshev_Theorem/Proof_2 | [
"Bertrand-Chebyshev Theorem",
"Number Theory"
] | [
"Definition:Prime Number"
] | [
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Prime Factor",
"Definition:Prime Number",
"Definition:Prime-Counting Function",
"Definition:Prime Number",
"Definition:Composite Number",
"Definition:Prime Number",
"Definition:Contradiction",
"Definitio... |
proofwiki-10267 | Sequence of Binomial Coefficients is Strictly Increasing to Half Upper Index | Let $n \in \Z_{>0}$ be a strictly positive integer.
Let $\dbinom n k$ be the binomial coefficient of $n$ over $k$ for a positive integer $k \in \Z_{\ge 0}$.
Let $S_n = \sequence {x_k}$ be the sequence defined as:
:$x_k = \dbinom n k$
Then $S_n$ is strictly increasing exactly where $0 \le k < \dfrac n 2$. | When $k \ge 0$, we have:
{{begin-eqn}}
{{eqn | l = \binom n {k + 1}
| r = \frac {n!} {\paren {k + 1}! \paren {n - k - 1}!}
| c = {{Defof|Binomial Coefficient}}
}}
{{eqn | r = \frac {n - k} {n - k} \frac {n!} {\paren {k + 1}! \paren {n - k - 1}!}
| c =
}}
{{eqn | r = \frac {n - k} {\paren {k + 1} \par... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|strictly positive integer]].
Let $\dbinom n k$ be the [[Definition:Binomial Coefficient|binomial coefficient]] of $n$ over $k$ for a [[Definition:Positive Integer|positive integer]] $k \in \Z_{\ge 0}$.
Let $S_n = \sequence {x_k}$ be the [[Definition:Sequ... | When $k \ge 0$, we have:
{{begin-eqn}}
{{eqn | l = \binom n {k + 1}
| r = \frac {n!} {\paren {k + 1}! \paren {n - k - 1}!}
| c = {{Defof|Binomial Coefficient}}
}}
{{eqn | r = \frac {n - k} {n - k} \frac {n!} {\paren {k + 1}! \paren {n - k - 1}!}
| c =
}}
{{eqn | r = \frac {n - k} {\paren {k + 1} \pa... | Sequence of Binomial Coefficients is Strictly Increasing to Half Upper Index | https://proofwiki.org/wiki/Sequence_of_Binomial_Coefficients_is_Strictly_Increasing_to_Half_Upper_Index | https://proofwiki.org/wiki/Sequence_of_Binomial_Coefficients_is_Strictly_Increasing_to_Half_Upper_Index | [
"Binomial Coefficients"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Binomial Coefficient",
"Definition:Positive/Integer",
"Definition:Sequence",
"Definition:Strictly Increasing/Sequence"
] | [
"Definition:Factorial",
"Definition:Factorial",
"Definition:Strictly Increasing/Sequence",
"Category:Binomial Coefficients"
] |
proofwiki-10268 | Isometry between Metric Spaces is Continuous | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.
Let $\phi: M_1 \to M_2$ be an isometry.
Then $\phi: M_1 \to M_2$ is a continuous mapping. | Let $a \in A_1$.
Let $\epsilon \in \R_{>0}$.
Let $\delta = \epsilon$.
Then:
{{begin-eqn}}
{{eqn | l = \map {d_1} {a, y}
| o = <
| r = \delta
| c = for some $y \in A_1$
}}
{{eqn | ll= \leadsto
| l = \map {d_2} {\map \phi a, \map \phi y}
| o = <
| r = \delta
| c = as $\map {d_2} ... | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be [[Definition:Metric Space|metric spaces]].
Let $\phi: M_1 \to M_2$ be an [[Definition:Isometry (Metric Spaces)|isometry]].
Then $\phi: M_1 \to M_2$ is a [[Definition:Continuous on Metric Space|continuous mapping]]. | Let $a \in A_1$.
Let $\epsilon \in \R_{>0}$.
Let $\delta = \epsilon$.
Then:
{{begin-eqn}}
{{eqn | l = \map {d_1} {a, y}
| o = <
| r = \delta
| c = for some $y \in A_1$
}}
{{eqn | ll= \leadsto
| l = \map {d_2} {\map \phi a, \map \phi y}
| o = <
| r = \delta
| c = as $\map {... | Isometry between Metric Spaces is Continuous | https://proofwiki.org/wiki/Isometry_between_Metric_Spaces_is_Continuous | https://proofwiki.org/wiki/Isometry_between_Metric_Spaces_is_Continuous | [
"Isometry between Metric Spaces is Continuous",
"Isometries (Metric Spaces)"
] | [
"Definition:Metric Space",
"Definition:Isometry (Metric Spaces)",
"Definition:Continuous Mapping (Metric Space)/Space"
] | [
"Definition:Continuous Mapping (Metric Space)/Point",
"Definition:Continuous Mapping (Metric Space)/Space"
] |
proofwiki-10269 | Equivalence of Definitions of Topologically Equivalent Metrics | {{TFAE|def = Topologically Equivalent Metrics}}
Let $M_1 = \struct {A, d_1}$ and $M_2 = \struct {A, d_2}$ be metric spaces on the same underlying set $A$. | === Definition 1 implies Definition 2 ===
Let $d_1$ and $d_2$ be topologically equivalent by definition 1.
Let $\struct {B, d}$ and $\struct {C, d'}$ be metric spaces:
Let $f: B \to A$ and $g: A \to C$ be mappings.
Then by definition:
:$f$ is $\tuple {d, d_1}$-continuous {{iff}} $f$ is $\tuple {d, d_2}$-continuous
:$g$... | {{TFAE|def = Topologically Equivalent Metrics}}
Let $M_1 = \struct {A, d_1}$ and $M_2 = \struct {A, d_2}$ be [[Definition:Metric Space|metric spaces]] on the same [[Definition:Underlying Set of Metric Space|underlying set]] $A$. | === Definition 1 implies Definition 2 ===
Let $d_1$ and $d_2$ be [[Definition:Topologically Equivalent Metrics/Definition 1|topologically equivalent by definition 1]].
Let $\struct {B, d}$ and $\struct {C, d'}$ be [[Definition:Metric Space|metric spaces]]:
Let $f: B \to A$ and $g: A \to C$ be [[Definition:Mapping|m... | Equivalence of Definitions of Topologically Equivalent Metrics | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Topologically_Equivalent_Metrics | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Topologically_Equivalent_Metrics | [
"Topologically Equivalent Metrics"
] | [
"Definition:Metric Space",
"Definition:Underlying Set/Metric Space"
] | [
"Definition:Topologically Equivalent Metrics/Definition 1",
"Definition:Metric Space",
"Definition:Mapping",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Continuous Mapping (Metric Space)",
"... |
proofwiki-10270 | Isometric Metric Spaces are Homeomorphic | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.
Let $M_1$ and $M_2$ be isometric.
Then $M_1$ and $M_2$ are homeomorphic. | By the definition of an isometry, there exists a bijection $f: A_1 \to A_2$ such that:
:$\forall a, b \in A_1: \map {d_1} {a, b} = \map {d_2} {\map \phi a, \map \phi b}$
From Isometry of Metric Spaces is Homeomorphism, $f$ is a homeomorphism from $M_1$ to $M_2$.
The result follows by definition of homeomorphic metric s... | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be [[Definition:Metric Space|metric spaces]].
Let $M_1$ and $M_2$ be [[Definition:Isometric Metric Spaces|isometric]].
Then $M_1$ and $M_2$ are [[Definition:Homeomorphic Metric Spaces|homeomorphic]]. | By the definition of an [[Definition:Isometry (Metric Spaces)|isometry]], there exists a [[Definition:Bijection|bijection]] $f: A_1 \to A_2$ such that:
:$\forall a, b \in A_1: \map {d_1} {a, b} = \map {d_2} {\map \phi a, \map \phi b}$
From [[Isometry of Metric Spaces is Homeomorphism]], $f$ is a [[Definition:Homeomorp... | Isometric Metric Spaces are Homeomorphic | https://proofwiki.org/wiki/Isometric_Metric_Spaces_are_Homeomorphic | https://proofwiki.org/wiki/Isometric_Metric_Spaces_are_Homeomorphic | [
"Isometries (Metric Spaces)",
"Homeomorphisms (Metric Spaces)"
] | [
"Definition:Metric Space",
"Definition:Isometry (Metric Spaces)",
"Definition:Homeomorphism/Metric Spaces"
] | [
"Definition:Isometry (Metric Spaces)",
"Definition:Bijection",
"Isometry of Metric Spaces is Homeomorphism",
"Definition:Homeomorphism/Metric Spaces",
"Definition:Homeomorphism/Metric Spaces"
] |
proofwiki-10271 | Homeomorphic Metric Spaces are not necessarily Isometric | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.
Let $M_1$ and $M_2$ be homeomorphic.
Then it is not necessarily the case that $M_1$ and $M_2$ are isometric. | Consider the spaces $\struct {\openint 0 4, d}$ and $\struct {\openint 0 1, d}$, where $d$ is the Euclidean Metric.
By Open Real Intervals are Homeomorphic, they are homeomorphic.
Let $\phi: \openint 0 4 \to \openint 0 1$ be a mapping.
Then:
{{begin-eqn}}
{{eqn | l = \map d {\map \phi 1, \map \phi 3}
| o = \le
... | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be [[Definition:Metric Space|metric spaces]].
Let $M_1$ and $M_2$ be [[Definition:Homeomorphic Metric Spaces|homeomorphic]].
Then it is not necessarily the case that $M_1$ and $M_2$ are [[Definition:Isometric Metric Spaces|isometric]]. | Consider the spaces $\struct {\openint 0 4, d}$ and $\struct {\openint 0 1, d}$, where $d$ is the [[Definition:Euclidean Metric on Real Number Line|Euclidean Metric]].
By [[Open Real Intervals are Homeomorphic]], they are [[Definition:Homeomorphic Metric Spaces|homeomorphic]].
Let $\phi: \openint 0 4 \to \openint 0 ... | Homeomorphic Metric Spaces are not necessarily Isometric | https://proofwiki.org/wiki/Homeomorphic_Metric_Spaces_are_not_necessarily_Isometric | https://proofwiki.org/wiki/Homeomorphic_Metric_Spaces_are_not_necessarily_Isometric | [
"Isometries (Metric Spaces)",
"Homeomorphisms (Metric Spaces)"
] | [
"Definition:Metric Space",
"Definition:Homeomorphism/Metric Spaces",
"Definition:Isometry (Metric Spaces)"
] | [
"Definition:Euclidean Metric/Real Number Line",
"Open Real Intervals are Homeomorphic",
"Definition:Homeomorphism/Metric Spaces",
"Definition:Isometry (Metric Spaces)",
"Definition:Isometry (Metric Spaces)"
] |
proofwiki-10272 | Identity Mapping between Metrics separated by Scale Factor is Continuous | Let $M_1 = \struct {A, d_1}$ and $M_2 = \struct {A, d_2}$ be metric spaces on the same underlying set $A$.
Let $d_1$ and $d_2$ be such that:
:$\forall x, y \in A: \map {d_2} {x, y} \le K \map {d_2} {x, y}$
Let $I_A: A \to A$ be the identity mapping on $A$.
Then $I_A$ is continuous from $M_1$ to $M_2$. | Let $\epsilon \in \R_{>0}$.
Let $a \in A$.
Set $\delta = \dfrac \epsilon K$.
Then:
{{begin-eqn}}
{{eqn | l = \map {d_1} {x, a}
| o = <
| r = \delta
| c =
}}
{{eqn | ll= \leadsto
| l = \map {d_2} {\map {I_A} x, \map {I_A} a}
| o = \le
| r = K \map {d_1} {x, a}
| c =
}}
{{eqn |... | Let $M_1 = \struct {A, d_1}$ and $M_2 = \struct {A, d_2}$ be [[Definition:Metric Space|metric spaces]] on the same [[Definition:Underlying Set of Metric Space|underlying set]] $A$.
Let $d_1$ and $d_2$ be such that:
:$\forall x, y \in A: \map {d_2} {x, y} \le K \map {d_2} {x, y}$
Let $I_A: A \to A$ be the [[Definitio... | Let $\epsilon \in \R_{>0}$.
Let $a \in A$.
Set $\delta = \dfrac \epsilon K$.
Then:
{{begin-eqn}}
{{eqn | l = \map {d_1} {x, a}
| o = <
| r = \delta
| c =
}}
{{eqn | ll= \leadsto
| l = \map {d_2} {\map {I_A} x, \map {I_A} a}
| o = \le
| r = K \map {d_1} {x, a}
| c =
}}
{{e... | Identity Mapping between Metrics separated by Scale Factor is Continuous | https://proofwiki.org/wiki/Identity_Mapping_between_Metrics_separated_by_Scale_Factor_is_Continuous | https://proofwiki.org/wiki/Identity_Mapping_between_Metrics_separated_by_Scale_Factor_is_Continuous | [
"Identity Mappings",
"Continuous Mappings on Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Underlying Set/Metric Space",
"Definition:Identity Mapping",
"Definition:Continuous Mapping (Metric Space)/Space"
] | [
"Definition:Continuous Mapping (Metric Space)/Point",
"Definition:Continuous Mapping (Metric Space)/Space"
] |
proofwiki-10273 | Metric Spaces on Topologically Equivalent Metrics on same Underlying Set are Homeomorphic | Let $M_1 = \struct {A, d_1}$ and $M_2 = \struct {A, d_2}$ be metric spaces on the same underlying set $A$.
Let $d_1$ and $d_2$ be topologically equivalent.
Then $M_1$ and $M_2$ are homeomorphic. | By definition, $d_1$ and $d_2$ are topologically equivalent on $A$ {{iff}}:
:For all metric spaces $\struct {B, d}$ and $\struct {C, d'}$:
:For all mappings $f: B \to A$ and $g: A \to C$:
::$(1): \quad f$ is $\tuple {d, d_1}$-continuous {{iff}} $f$ is $\tuple {d, d_2}$-continuous
::$(2): \quad g$ is $\tuple {d_1, d'}$-... | Let $M_1 = \struct {A, d_1}$ and $M_2 = \struct {A, d_2}$ be [[Definition:Metric Space|metric spaces]] on the same [[Definition:Underlying Set of Metric Space|underlying set]] $A$.
Let $d_1$ and $d_2$ be [[Definition:Topologically Equivalent Metrics|topologically equivalent]].
Then $M_1$ and $M_2$ are [[Definition:H... | By definition, $d_1$ and $d_2$ are [[Definition:Topologically Equivalent Metrics|topologically equivalent]] on $A$ {{iff}}:
:For all [[Definition:Metric Space|metric spaces]] $\struct {B, d}$ and $\struct {C, d'}$:
:For all [[Definition:Mapping|mappings]] $f: B \to A$ and $g: A \to C$:
::$(1): \quad f$ is [[Definition... | Metric Spaces on Topologically Equivalent Metrics on same Underlying Set are Homeomorphic | https://proofwiki.org/wiki/Metric_Spaces_on_Topologically_Equivalent_Metrics_on_same_Underlying_Set_are_Homeomorphic | https://proofwiki.org/wiki/Metric_Spaces_on_Topologically_Equivalent_Metrics_on_same_Underlying_Set_are_Homeomorphic | [
"Topologically Equivalent Metrics",
"Homeomorphisms (Metric Spaces)"
] | [
"Definition:Metric Space",
"Definition:Underlying Set/Metric Space",
"Definition:Topologically Equivalent Metrics",
"Definition:Homeomorphism/Metric Spaces"
] | [
"Definition:Topologically Equivalent Metrics",
"Definition:Metric Space",
"Definition:Mapping",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Ho... |
proofwiki-10274 | Real Vector Space under Chebyshev Distance is Homeomorphic to that under Euclidean Metric | Let $\R^n$ be an $n$-dimensional real vector space.
Let $d_\infty: \R^n \times \R^n \to \R$ be the Chebyshev distance on $\R^n$.
Let $d_2: \R^n \times \R^n \to \R$ be the Euclidean metric on $\R^n$.
Let $M_1 = \struct {\R^n, d_\infty}$ and $M_2 = \struct {\R^n, d_2}$ be the corresponding metric spaces.
Then $M_1$ and $... | From Relation between $p$-Product Metric and Chebyshev Distance on Real Vector Space:
:$\forall x, y \in \R^n: \map {d_\infty} {x, y} \le \map {d_p} {x, y} \le n^{1 / p} \map {d_\infty} {x, y}$
The Euclidean metric $d_2$ is a special case of the $p$-product metric $d_p$ for $p = 2$.
It follows by definition that $d_\in... | Let $\R^n$ be an [[Definition:Dimension of Vector Space|$n$-dimensional]] [[Definition:Real Vector Space|real vector space]].
Let $d_\infty: \R^n \times \R^n \to \R$ be the [[Definition:Chebyshev Distance on Real Vector Space|Chebyshev distance on $\R^n$]].
Let $d_2: \R^n \times \R^n \to \R$ be the [[Definition:Eucli... | From [[Relation between P-Product Metric and Chebyshev Distance on Real Vector Space|Relation between $p$-Product Metric and Chebyshev Distance on Real Vector Space]]:
:$\forall x, y \in \R^n: \map {d_\infty} {x, y} \le \map {d_p} {x, y} \le n^{1 / p} \map {d_\infty} {x, y}$
The [[Definition:Euclidean Metric on Real V... | Real Vector Space under Chebyshev Distance is Homeomorphic to that under Euclidean Metric | https://proofwiki.org/wiki/Real_Vector_Space_under_Chebyshev_Distance_is_Homeomorphic_to_that_under_Euclidean_Metric | https://proofwiki.org/wiki/Real_Vector_Space_under_Chebyshev_Distance_is_Homeomorphic_to_that_under_Euclidean_Metric | [
"Homeomorphisms (Metric Spaces)",
"Chebyshev Distance",
"Euclidean Metric"
] | [
"Definition:Dimension of Vector Space",
"Definition:Real Vector Space",
"Definition:Chebyshev Distance/Real Vector Space",
"Definition:Euclidean Metric/Real Vector Space",
"Definition:Metric Space",
"Definition:Homeomorphism/Metric Spaces"
] | [
"Relation between P-Product Metric and Chebyshev Distance on Real Vector Space",
"Definition:Euclidean Metric/Real Vector Space",
"Definition:P-Product Metric/Real Vector Space",
"Definition:Lipschitz Equivalence/Metrics",
"Lipschitz Equivalent Metrics are Topologically Equivalent",
"Definition:Topologica... |
proofwiki-10275 | Equivalence of Definitions of Homeomorphism between Metric Spaces | {{TFAE|def = Homeomorphism (Metric Spaces)}}
Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.
Let $f: A_1 \to A_2$ be a bijection. | In order to prove the assertion it is sufficient to prove that the conditions for homeomorphism in definitions $2$ to $4$ are necessary and sufficient conditions for $f$ and $f^{-1}$ to be continuous on $M_1$ and $M_2$ respectively. | {{TFAE|def = Homeomorphism (Metric Spaces)}}
Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be [[Definition:Metric Space|metric spaces]].
Let $f: A_1 \to A_2$ be a [[Definition:Bijection|bijection]]. | In order to prove the assertion it is sufficient to prove that the conditions for [[Definition:Homeomorphism (Metric Spaces)|homeomorphism]] in definitions $2$ to $4$ are [[Definition:Necessary and Sufficient|necessary and sufficient conditions]] for $f$ and $f^{-1}$ to be [[Definition:Continuous on Metric Space|contin... | Equivalence of Definitions of Homeomorphism between Metric Spaces | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Homeomorphism_between_Metric_Spaces | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Homeomorphism_between_Metric_Spaces | [
"Homeomorphism (Metric Spaces)"
] | [
"Definition:Metric Space",
"Definition:Bijection"
] | [
"Definition:Homeomorphism/Metric Spaces",
"Definition:Biconditional/Semantics of Biconditional/Necessary and Sufficient",
"Definition:Continuous Mapping (Metric Space)/Space",
"Definition:Continuous Mapping (Metric Space)/Space",
"Definition:Continuous Mapping (Metric Space)/Space",
"Definition:Continuous... |
proofwiki-10276 | Liouville's Theorem (Number Theory) | Let $x$ be an irrational number that is algebraic of degree $n$.
Then there exists a constant $c > 0$ (which can depend on $x$) such that:
:$\size {x - \dfrac p q} \ge \dfrac c {q^n}$
for every pair $p, q \in \Z$ with $q \ne 0$. | Let $r_1, r_2, \ldots, r_k$ be the rational roots of a polynomial $P$ of degree $n$ that has $x$ as a root.
Since $x$ is irrational, it does not equal any $r_i$.
Let $c_1 > 0$ be the minimum of $\size {x - r_i}$.
If there are no $r_i$, let $c_1 = 1$.
Now let $\alpha = \dfrac p q$ where $\alpha \notin \set {r_1, \ldots... | Let $x$ be an [[Definition:Irrational Number|irrational number]] that is [[Definition:Algebraic Number|algebraic]] of [[Definition:Degree of Algebraic Number|degree]] $n$.
Then there exists a constant $c > 0$ (which can depend on $x$) such that:
:$\size {x - \dfrac p q} \ge \dfrac c {q^n}$
for every pair $p, q \in \... | Let $r_1, r_2, \ldots, r_k$ be the [[Definition:Rational Number|rational]] [[Definition:Root of Polynomial|roots]] of a [[Definition:Polynomial (Analysis)|polynomial]] $P$ of [[Definition:Degree of Polynomial|degree]] $n$ that has $x$ as a [[Definition:Root of Polynomial|root]].
Since $x$ is [[Definition:Irrational Nu... | Liouville's Theorem (Number Theory) | https://proofwiki.org/wiki/Liouville's_Theorem_(Number_Theory) | https://proofwiki.org/wiki/Liouville's_Theorem_(Number_Theory) | [
"Liouville's Theorem (Number Theory)",
"Number Theory",
"Transcendental Number Theory",
"Liouville's Theorem"
] | [
"Definition:Irrational Number",
"Definition:Algebraic Number",
"Definition:Algebraic Number/Degree"
] | [
"Definition:Rational Number",
"Definition:Root of Polynomial",
"Definition:Polynomial",
"Definition:Degree of Polynomial",
"Definition:Root of Polynomial",
"Definition:Irrational Number",
"Difference of Two Powers",
"Reverse Triangle Inequality",
"Triangle Inequality",
"Sum of Geometric Sequence"
... |
proofwiki-10277 | Liouville Numbers are Irrational | Liouville numbers are irrational. | Let $x$ be a Liouville number.
{{AimForCont}} $x$ were rational, that is:
: $x = \dfrac a b$
with $a, b \in \Z$ and $b > 0$.
By definition of a Liouville number, for all $n \in \N$, there exist $p,q \in \Z$ (which may depend on $n$) with $q > 1$ such that:
:$0 < \size {x - \dfrac p q} < \dfrac 1 {q^n}$
Let $n$ be a po... | [[Definition:Liouville Number|Liouville numbers]] are [[Definition:Irrational Number|irrational]]. | Let $x$ be a [[Definition:Liouville Number|Liouville number]].
{{AimForCont}} $x$ were [[Definition:Rational Number|rational]], that is:
: $x = \dfrac a b$
with $a, b \in \Z$ and $b > 0$.
By definition of a [[Definition:Liouville Number|Liouville number]], for all $n \in \N$, there exist $p,q \in \Z$ (which may depe... | Liouville Numbers are Irrational | https://proofwiki.org/wiki/Liouville_Numbers_are_Irrational | https://proofwiki.org/wiki/Liouville_Numbers_are_Irrational | [
"Number Theory"
] | [
"Definition:Liouville Number",
"Definition:Irrational Number"
] | [
"Definition:Liouville Number",
"Definition:Rational Number",
"Definition:Liouville Number",
"Definition:Positive/Integer",
"Definition:Integer",
"Definition:Positive/Integer",
"Definition:Irrational Number",
"Category:Number Theory"
] |
proofwiki-10278 | Cardinality of Power Set of Natural Numbers Equals Cardinality of Real Numbers | The cardinality of the power set of the natural numbers is equal to the cardinality of the real numbers.
:$\card {2^\N} = \card \R$ | {{refactor|level = basic|What is being stated is covered in passing by the below}}
{{mergeto|Power Set of Natural Numbers has Cardinality of Continuum}}
This is a direct corollary of:
:Power Set of Natural Numbers has Cardinality of Continuum
:Cardinality of Real Numbers is Cardinality of Continuum.
{{qed}} | The [[Definition:Cardinality|cardinality]] of the [[Definition:Power Set|power set]] of the [[Definition:Natural Numbers|natural numbers]] is equal to the [[Definition:Cardinality|cardinality]] of the [[Definition:Real Number|real numbers]].
:$\card {2^\N} = \card \R$ | {{refactor|level = basic|What is being stated is covered in passing by the below}}
{{mergeto|Power Set of Natural Numbers has Cardinality of Continuum}}
This is a direct corollary of:
:[[Power Set of Natural Numbers has Cardinality of Continuum]]
:[[Cardinality of Real Numbers is Cardinality of Continuum]].
{{qed}} | Cardinality of Power Set of Natural Numbers Equals Cardinality of Real Numbers | https://proofwiki.org/wiki/Cardinality_of_Power_Set_of_Natural_Numbers_Equals_Cardinality_of_Real_Numbers | https://proofwiki.org/wiki/Cardinality_of_Power_Set_of_Natural_Numbers_Equals_Cardinality_of_Real_Numbers | [
"Cardinality of Power Set",
"Cardinality",
"Power Set",
"Natural Numbers",
"Infinite Sets",
"Exponentiation of Cardinals",
"Cardinal Arithmetic"
] | [
"Definition:Cardinality",
"Definition:Power Set",
"Definition:Natural Numbers",
"Definition:Cardinality",
"Definition:Real Number"
] | [
"Power Set of Natural Numbers has Cardinality of Continuum",
"Cardinality of Real Numbers is Cardinality of Continuum"
] |
proofwiki-10279 | Liouville's Constant is Transcendental | Liouville's constant:
{{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 1}^\infty \dfrac 1 {10^{n!} }
| r = \frac 1 {10^1} + \frac 1 {10^2} + \frac 1 {10^6} + \frac 1 {10^{24} } + \cdots
| c =
}}
{{eqn | r = 0.11000 \, 10000 \, 00000 \, 00000 \, 00010 \, 00 \ldots
| c =
}}
{{end-eqn}}
is transcendental. | Let $q = 10^{n!}$ and write:
:$\ds L = \frac p q + \sum_{k \mathop = n + 1}^\infty \frac 1 {10^{k!} }$
for some suitable $p \in \Z$.
Then:
{{begin-eqn}}
{{eqn | l = \size {L - \frac p q}
| r = \sum_{k \mathop = n + 1}^\infty \frac 1 {10^{k!} }
| c =
}}
{{eqn | r = \frac 2 {10^{\paren {n + 1}!} }
| o... | [[Definition:Liouville's Constant|Liouville's constant]]:
{{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 1}^\infty \dfrac 1 {10^{n!} }
| r = \frac 1 {10^1} + \frac 1 {10^2} + \frac 1 {10^6} + \frac 1 {10^{24} } + \cdots
| c =
}}
{{eqn | r = 0.11000 \, 10000 \, 00000 \, 00000 \, 00010 \, 00 \ldots
| c ... | Let $q = 10^{n!}$ and write:
:$\ds L = \frac p q + \sum_{k \mathop = n + 1}^\infty \frac 1 {10^{k!} }$
for some suitable $p \in \Z$.
Then:
{{begin-eqn}}
{{eqn | l = \size {L - \frac p q}
| r = \sum_{k \mathop = n + 1}^\infty \frac 1 {10^{k!} }
| c =
}}
{{eqn | r = \frac 2 {10^{\paren {n + 1}!} }
... | Liouville's Constant is Transcendental | https://proofwiki.org/wiki/Liouville's_Constant_is_Transcendental | https://proofwiki.org/wiki/Liouville's_Constant_is_Transcendental | [
"Transcendental Number Theory",
"Transcendental Numbers"
] | [
"Definition:Liouville's Constant",
"Definition:Transcendental Number"
] | [
"Definition:Liouville Number",
"Liouville's Theorem (Number Theory)/Corollary",
"Definition:Transcendental Number"
] |
proofwiki-10280 | Isometry of Metric Spaces is Equivalence Relation | Let $M_1$ and $M_2$ be metric spaces.
Let $M_1 \sim M_2$ denote that $M_1$ and $M_2$ are isometric.
The relation $\sim$ is an equivalence relation. | Checking in turn each of the criteria for equivalence: | Let $M_1$ and $M_2$ be [[Definition:Metric Space|metric spaces]].
Let $M_1 \sim M_2$ denote that $M_1$ and $M_2$ are [[Definition:Isometric Metric Spaces|isometric]].
The relation $\sim$ is an [[Definition:Equivalence Relation|equivalence relation]]. | Checking in turn each of the criteria for [[Definition:Equivalence Relation|equivalence]]: | Isometry of Metric Spaces is Equivalence Relation | https://proofwiki.org/wiki/Isometry_of_Metric_Spaces_is_Equivalence_Relation | https://proofwiki.org/wiki/Isometry_of_Metric_Spaces_is_Equivalence_Relation | [
"Isometries (Metric Spaces)",
"Examples of Equivalence Relations"
] | [
"Definition:Metric Space",
"Definition:Isometry (Metric Spaces)",
"Definition:Equivalence Relation"
] | [
"Definition:Equivalence Relation",
"Definition:Equivalence Relation"
] |
proofwiki-10281 | Homeomorphism of Metric Spaces is Equivalence Relation | Let $M_1$ and $M_2$ be metric spaces.
Let $M_1 \sim M_2$ denote that $M_1$ and $M_2$ are homeomorphic.
The relation $\sim$ is an equivalence relation. | Checking in turn each of the criteria for equivalence: | Let $M_1$ and $M_2$ be [[Definition:Metric Space|metric spaces]].
Let $M_1 \sim M_2$ denote that $M_1$ and $M_2$ are [[Definition:Homeomorphic Metric Spaces|homeomorphic]].
The relation $\sim$ is an [[Definition:Equivalence Relation|equivalence relation]]. | Checking in turn each of the criteria for [[Definition:Equivalence Relation|equivalence]]: | Homeomorphism of Metric Spaces is Equivalence Relation | https://proofwiki.org/wiki/Homeomorphism_of_Metric_Spaces_is_Equivalence_Relation | https://proofwiki.org/wiki/Homeomorphism_of_Metric_Spaces_is_Equivalence_Relation | [
"Homeomorphisms (Metric Spaces)",
"Examples of Equivalence Relations"
] | [
"Definition:Metric Space",
"Definition:Homeomorphism/Metric Spaces",
"Definition:Equivalence Relation"
] | [
"Definition:Equivalence Relation",
"Definition:Equivalence Relation"
] |
proofwiki-10282 | Isometry of Metric Spaces is Homeomorphism | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.
Let $f: M_1 \to M_2$ be an isometry.
Then $f$ is a homeomorphism from $M_1$ to $M_2$. | By the definition of an isometry, $f$ is a bijection $f: A_1 \to A_2$ such that:
:$\forall a, b \in A_1: \map {d_1} {a, b} = \map {d_2} {\map \phi a, \map \phi b}$
By Isometry between Metric Spaces is Continuous, $f$ is a continuous mapping from $M_1$ to $M_2$.
By {{Corollary|Isometry between Metric Spaces is Continuou... | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be [[Definition:Metric Space|metric spaces]].
Let $f: M_1 \to M_2$ be an [[Definition:Isometry (Metric Spaces)|isometry]].
Then $f$ is a [[Definition:Homeomorphism (Metric Spaces)|homeomorphism]] from $M_1$ to $M_2$. | By the definition of an [[Definition:Isometry (Metric Spaces)|isometry]], $f$ is a [[Definition:Bijection|bijection]] $f: A_1 \to A_2$ such that:
:$\forall a, b \in A_1: \map {d_1} {a, b} = \map {d_2} {\map \phi a, \map \phi b}$
By [[Isometry between Metric Spaces is Continuous]], $f$ is a [[Definition:Continuous on ... | Isometry of Metric Spaces is Homeomorphism | https://proofwiki.org/wiki/Isometry_of_Metric_Spaces_is_Homeomorphism | https://proofwiki.org/wiki/Isometry_of_Metric_Spaces_is_Homeomorphism | [
"Isometries (Metric Spaces)",
"Homeomorphisms (Metric Spaces)"
] | [
"Definition:Metric Space",
"Definition:Isometry (Metric Spaces)",
"Definition:Homeomorphism/Metric Spaces"
] | [
"Definition:Isometry (Metric Spaces)",
"Definition:Bijection",
"Isometry between Metric Spaces is Continuous",
"Definition:Continuous Mapping (Metric Space)/Space",
"Definition:Continuous Mapping (Metric Space)/Space",
"Definition:Homeomorphism/Metric Spaces",
"Category:Isometries (Metric Spaces)",
"C... |
proofwiki-10283 | Equivalence Class of Isometries is Subset of Equivalence Class of Homeomorphisms | Let $\CC_I$ be an equivalence class of isometries on the set of metric spaces.
Then $\CC_I$ is a subset of an equivalence class of homeomorphisms on the set of metric spaces. | From Isometry of Metric Spaces is Equivalence Relation, all isometries can be partitioned into equivalence classes.
From Homeomorphism of Metric Spaces is Equivalence Relation, all homeomorphisms can be partitioned into equivalence classes.
Let $\CC_I$ be an equivalence class of isometries.
Let $f: M_1 \to M_2$ be an e... | Let $\CC_I$ be an [[Definition:Equivalence Class|equivalence class]] of [[Definition:Isometry (Metric Spaces)|isometries]] on the [[Definition:Set|set]] of [[Definition:Metric Space|metric spaces]].
Then $\CC_I$ is a [[Definition:Subset|subset]] of an [[Definition:Equivalence Class|equivalence class]] of [[Definition:... | From [[Isometry of Metric Spaces is Equivalence Relation]], all [[Definition:Isometry (Metric Spaces)|isometries]] can be partitioned into [[Definition:Equivalence Class|equivalence classes]].
From [[Homeomorphism of Metric Spaces is Equivalence Relation]], all [[Definition:Homeomorphism (Metric Spaces)|homeomorphisms... | Equivalence Class of Isometries is Subset of Equivalence Class of Homeomorphisms | https://proofwiki.org/wiki/Equivalence_Class_of_Isometries_is_Subset_of_Equivalence_Class_of_Homeomorphisms | https://proofwiki.org/wiki/Equivalence_Class_of_Isometries_is_Subset_of_Equivalence_Class_of_Homeomorphisms | [
"Homeomorphisms (Metric Spaces)",
"Isometries (Metric Spaces)",
"Equivalence Classes"
] | [
"Definition:Equivalence Class",
"Definition:Isometry (Metric Spaces)",
"Definition:Set",
"Definition:Metric Space",
"Definition:Subset",
"Definition:Equivalence Class",
"Definition:Homeomorphism/Metric Spaces",
"Definition:Set",
"Definition:Metric Space"
] | [
"Isometry of Metric Spaces is Equivalence Relation",
"Definition:Isometry (Metric Spaces)",
"Definition:Equivalence Class",
"Homeomorphism of Metric Spaces is Equivalence Relation",
"Definition:Homeomorphism/Metric Spaces",
"Definition:Equivalence Class",
"Definition:Equivalence Class",
"Definition:Is... |
proofwiki-10284 | Equivalence of Local Uniform Convergence and Compact Convergence | Let $U \subseteq \C$ be an open set of the complex plane.
Let $f_n: U \to \C$ be a sequence of functions which converges pointwise to $f: U \to \C$.
Then $f_n$ converges to $f$ uniformly on all compact subsets of $U$ {{iff}} $f_n$ converges locally uniformly on $U$. | === Sufficient Condition ===
Suppose that $f_n$ converges to $f$ uniformly on all compact subsets of $U$.
Let $z \in U$.
Since $U$ is open, there is an open ball $\map {B_\epsilon} z$, around $z$, of radius $\epsilon$, with the closure of the ball, $\overline{\map {B_\epsilon} z}$, contained in $U$.
Since $\overline {\... | Let $U \subseteq \C$ be an [[Definition:Open Set (Complex Analysis)|open set]] of the [[Definition:Complex Plane|complex plane]].
Let $f_n: U \to \C$ be a [[Definition:Sequence|sequence]] of functions which [[Definition:Pointwise Convergence|converges pointwise]] to $f: U \to \C$.
Then $f_n$ [[Definition:Uniform Con... | === Sufficient Condition ===
Suppose that $f_n$ [[Definition:Uniform Convergence|converges to $f$ uniformly]] on all [[Definition:Compact Subset of Complex Plane|compact subsets]] of $U$.
Let $z \in U$.
Since $U$ is [[Definition:Open Set (Complex Analysis)|open]], there is an [[Definition:Open Ball|open ball]] $\map... | Equivalence of Local Uniform Convergence and Compact Convergence | https://proofwiki.org/wiki/Equivalence_of_Local_Uniform_Convergence_and_Compact_Convergence | https://proofwiki.org/wiki/Equivalence_of_Local_Uniform_Convergence_and_Compact_Convergence | [
"Complex Analysis",
"Uniform Convergence"
] | [
"Definition:Open Set/Complex Analysis",
"Definition:Complex Number/Complex Plane",
"Definition:Sequence",
"Definition:Pointwise Convergence",
"Definition:Uniform Convergence",
"Definition:Compact Space/Metric Space/Complex",
"Definition:Locally Uniform Convergence"
] | [
"Definition:Uniform Convergence",
"Definition:Compact Space/Metric Space/Complex",
"Definition:Open Set/Complex Analysis",
"Definition:Open Ball",
"Definition:Open Ball/Radius",
"Definition:Closed Ball",
"Definition:Closed Set/Complex Analysis",
"Definition:Bounded Metric Space/Complex",
"Definition... |
proofwiki-10285 | Frattini's Argument | Let $\struct {G, \circ}$ be a group.
Let $K$ be a finite normal subgroup of $G$, and $p$ a prime which divides the order of $K$.
Let $P$ be a Sylow $p$-subgroup of $K$, and $\map {N_G} P$ the normalizer of $P$ in $G$.
Then:
:$G = \map {N_G} P \circ K = K \circ \map {N_G} P$ | Let $g \in G$.
Since $K$ is normal in $G$, and $P \subset K$, the conjugate $g \circ P \circ g^{-1}$ of $P$ is also a subset of $K$.
From Inner Automorphism is Automorphism, $g \circ P \circ g^{-1}$ is a subgroup of $K$ of the same order as $P$.
Thus $g \circ P \circ g^{-1}$ is also a Sylow $p$-subgroup of $K$.
By Seco... | Let $\struct {G, \circ}$ be a [[Definition:Group|group]].
Let $K$ be a [[Definition:Finite Group|finite]] [[Definition:Normal Subgroup|normal subgroup]] of $G$, and $p$ a [[Definition:Prime Number|prime]] which [[Definition:Divisor of Integer|divides]] the [[Definition:Order of Structure|order]] of $K$.
Let $P$ be a ... | Let $g \in G$.
Since $K$ is [[Definition:Normal Subgroup|normal]] in $G$, and $P \subset K$, the [[Definition:Conjugate of Group Subset|conjugate]] $g \circ P \circ g^{-1}$ of $P$ is also a [[Definition:Subset|subset]] of $K$.
From [[Inner Automorphism is Automorphism]], $g \circ P \circ g^{-1}$ is a [[Definition:Sub... | Frattini's Argument | https://proofwiki.org/wiki/Frattini's_Argument | https://proofwiki.org/wiki/Frattini's_Argument | [
"Group Theory",
"Normalizers"
] | [
"Definition:Group",
"Definition:Finite Group",
"Definition:Normal Subgroup",
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer",
"Definition:Order of Structure",
"Definition:Sylow p-Subgroup",
"Definition:Normalizer"
] | [
"Definition:Normal Subgroup",
"Definition:Conjugate (Group Theory)/Subset",
"Definition:Subset",
"Inner Automorphism is Automorphism",
"Definition:Subgroup",
"Definition:Order of Structure",
"Definition:Sylow p-Subgroup",
"Second Sylow Theorem",
"Definition:Normalizer",
"Definition:Normal Subgroup... |
proofwiki-10286 | Sum of Integrals on Complementary Sets | Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $A, E \in \Sigma$ with $A \subseteq E$.
Let $f$ be a $\mu$-integrable function on $X$.
Then
:$\ds \int_E f \rd \mu = \int_A f \rd \mu + \int_{E \mathop \setminus A} f \rd \mu$ | Let $\chi_E$ be the characteristic function of $E$.
Because $A$ and $E \setminus A$ are disjoint:
:$A \cap \paren {E \setminus A} = \O$
By Characteristic Function of Union:
:$\chi_E = \chi_A + \chi_{E \mathop \setminus A}$
Integrating $f$ over $E$ gives:
{{begin-eqn}}
{{eqn | l = \int_E f \rd \mu
| r = \int \chi... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]].
Let $A, E \in \Sigma$ with $A \subseteq E$.
Let $f$ be a [[Definition:Measure-Integrable Function|$\mu$-integrable function]] on $X$.
Then
:$\ds \int_E f \rd \mu = \int_A f \rd \mu + \int_{E \mathop \setminus A} f \rd \mu$ | Let $\chi_E$ be the [[Definition:Characteristic Function of Set|characteristic function]] of $E$.
Because $A$ and $E \setminus A$ are [[Definition:Disjoint Sets|disjoint]]:
:$A \cap \paren {E \setminus A} = \O$
By [[Characteristic Function of Union/Variant 2|Characteristic Function of Union]]:
:$\chi_E = \chi_A + \ch... | Sum of Integrals on Complementary Sets | https://proofwiki.org/wiki/Sum_of_Integrals_on_Complementary_Sets | https://proofwiki.org/wiki/Sum_of_Integrals_on_Complementary_Sets | [
"Integrals of Measure-Integrable Functions"
] | [
"Definition:Measure Space",
"Definition:Integrable Function/Measure Space"
] | [
"Definition:Characteristic Function (Set Theory)/Set",
"Definition:Disjoint Sets",
"Characteristic Function of Union/Variant 2",
"Definition:Integral of Measure-Integrable Function",
"Pointwise Operation on Distributive Structure is Distributive",
"Integral of Integrable Function is Additive"
] |
proofwiki-10287 | Closure of Pointwise Operation on Algebraic Structure | Let $S$ be a set such that $S \ne \O$.
Let $\struct {T, \circ}$ be an algebraic structure.
Let $T^S$ be the set of all mappings from $S$ to $T$.
Let $f, g \in T^S$, that is, let $f: S \to T$ and $g: S \to T$ be mappings.
Let $\oplus: T^S \to T^S$ be the pointwise operation on $T^S$ induced by $\circ$.
Then $\oplus$ is ... | === Necessary Condition ===
Let $\struct {T, \circ}$ be closed.
Let $x \in S$ be arbitrary.
Then:
{{begin-eqn}}
{{eqn | q = \forall f, g \in T^S
| l = \map {f \oplus g} x
| r = \map f x \circ \map g x
| c = {{Defof|Pointwise Operation}}
}}
{{eqn | o = \in
| r = T
| c = as $\struct {T, \cir... | Let $S$ be a [[Definition:Set|set]] such that $S \ne \O$.
Let $\struct {T, \circ}$ be an [[Definition:Algebraic Structure|algebraic structure]].
Let $T^S$ be the [[Definition:Set of All Mappings|set of all mappings]] from $S$ to $T$.
Let $f, g \in T^S$, that is, let $f: S \to T$ and $g: S \to T$ be [[Definition:Mapp... | === Necessary Condition ===
Let $\struct {T, \circ}$ be [[Definition:Closed Algebraic Structure|closed]].
Let $x \in S$ be arbitrary.
Then:
{{begin-eqn}}
{{eqn | q = \forall f, g \in T^S
| l = \map {f \oplus g} x
| r = \map f x \circ \map g x
| c = {{Defof|Pointwise Operation}}
}}
{{eqn | o = \in
... | Closure of Pointwise Operation on Algebraic Structure | https://proofwiki.org/wiki/Closure_of_Pointwise_Operation_on_Algebraic_Structure | https://proofwiki.org/wiki/Closure_of_Pointwise_Operation_on_Algebraic_Structure | [
"Pointwise Operations"
] | [
"Definition:Set",
"Definition:Algebraic Structure",
"Definition:Set of All Mappings",
"Definition:Mapping",
"Definition:Pointwise Operation",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Closure (Abstract Algebra)/Algebraic Structure"
] | [
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"... |
proofwiki-10288 | Pointwise Operation on Distributive Structure is Distributive | Let $S$ be a set.
Let $\struct {T, +, \circ}$ be an algebraic structure with two operations $+$ and $\circ$.
Let $T^S$ be the set of all mappings from $S$ to $T$.
Let $\struct {T^S, +', \circ'}$ be the structure induced on $T^S$ by $+$ and $\circ$.
Let $\circ$ be distributive over $+$ in $T$.
Then $\circ'$ is distribut... | Let $f, g, h: S \to T$ be elements of $T^S$.
Suppose $S$ is the empty set.
Suppose $T^S$ is the set of all mappings from the empty set, $S$, to $T$.
Suppose $\struct {T^S, +', \circ'}$ is the structure induced on $T^S$ by $+$ and $\circ$.
Suppose $\circ$ is distributive over $+$ in $T$.
{{Finish|Prove that the statemen... | Let $S$ be a [[Definition:Set|set]].
Let $\struct {T, +, \circ}$ be an [[Definition:Algebraic Structure|algebraic structure]] with two [[Definition:Binary Operation|operations]] $+$ and $\circ$.
Let $T^S$ be the [[Definition:Set of All Mappings|set of all mappings]] from $S$ to $T$.
Let $\struct {T^S, +', \circ'}$ b... | Let $f, g, h: S \to T$ be elements of $T^S$.
Suppose $S$ is the [[Definition:Empty Set|empty set]].
Suppose $T^S$ is the [[Definition:Set of All Mappings|set of all mappings]] from the [[Definition:Empty Set|empty set]], $S$, to $T$.
Suppose $\struct {T^S, +', \circ'}$ is the [[Definition:Induced Structure|structure... | Pointwise Operation on Distributive Structure is Distributive | https://proofwiki.org/wiki/Pointwise_Operation_on_Distributive_Structure_is_Distributive | https://proofwiki.org/wiki/Pointwise_Operation_on_Distributive_Structure_is_Distributive | [
"Pointwise Operations",
"Distributive Operations"
] | [
"Definition:Set",
"Definition:Algebraic Structure",
"Definition:Operation/Binary Operation",
"Definition:Set of All Mappings",
"Definition:Pointwise Operation/Induced Structure",
"Definition:Distributive Operation",
"Definition:Distributive Operation"
] | [
"Definition:Empty Set",
"Definition:Set of All Mappings",
"Definition:Empty Set",
"Definition:Pointwise Operation/Induced Structure",
"Definition:Distributive Operation",
"Definition:Distributive Operation",
"Definition:Non-Empty Set",
"Definition:Distributive Operation",
"Category:Pointwise Operati... |
proofwiki-10289 | Identity Mapping on Metric Space is Homeomorphism | Let $M = \struct {A, d}$ be a metric space.
The identity mapping $I_A: M \to M$ defined as:
:$\forall x \in A: \map {I_A} x = x$
is a homeomorphism. | We have Identity Mapping is Bijection.
We also have Identity Mapping is Continuous.
Hence, by definition, $I_T$ is a homeomorphism.
{{qed}}
Category:Homeomorphisms (Metric Spaces)
Category:Topologically Equivalent Metrics
Category:Identity Mappings
mxwnp6kgls3opibt01kued5od1f5hw8 | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
The [[Definition:Identity Mapping|identity mapping]] $I_A: M \to M$ defined as:
:$\forall x \in A: \map {I_A} x = x$
is a [[Definition:Homeomorphism (Metric Spaces)|homeomorphism]]. | We have [[Identity Mapping is Bijection]].
We also have [[Identity Mapping is Continuous/Metric Space|Identity Mapping is Continuous]].
Hence, by definition, $I_T$ is a [[Definition:Homeomorphism (Metric Spaces)|homeomorphism]].
{{qed}}
[[Category:Homeomorphisms (Metric Spaces)]]
[[Category:Topologically Equivalent ... | Identity Mapping on Metric Space is Homeomorphism | https://proofwiki.org/wiki/Identity_Mapping_on_Metric_Space_is_Homeomorphism | https://proofwiki.org/wiki/Identity_Mapping_on_Metric_Space_is_Homeomorphism | [
"Homeomorphisms (Metric Spaces)",
"Topologically Equivalent Metrics",
"Identity Mappings"
] | [
"Definition:Metric Space",
"Definition:Identity Mapping",
"Definition:Homeomorphism/Metric Spaces"
] | [
"Identity Mapping is Bijection",
"Identity Mapping is Continuous/Metric Space",
"Definition:Homeomorphism/Metric Spaces",
"Category:Homeomorphisms (Metric Spaces)",
"Category:Topologically Equivalent Metrics",
"Category:Identity Mappings"
] |
proofwiki-10290 | Inverse of Homeomorphism between Metric Spaces is Homeomorphism | Let $M = \struct {A_1, d_1}, M_2 = \struct {A_2, d_2}$ be metric spaces.
Let $f: M_1 \to M_2$ be a homeomorphism.
Then $f^{-1}: M_2 \to M_1$ is also a homeomorphism. | By definition, a homeomorphism is a bijection such that both $f$ and $f^{-1}$ are continuous.
As $f$ is a bijection then by Bijection iff Inverse is Bijection, so is $f^{-1}$.
So by definition $f^{-1}$ is a bijection such that both $f^{-1}$ and $\paren {f^{-1} }^{-1}$ are continuous.
The result follows from Inverse of ... | Let $M = \struct {A_1, d_1}, M_2 = \struct {A_2, d_2}$ be [[Definition:Metric Space|metric spaces]].
Let $f: M_1 \to M_2$ be a [[Definition:Homeomorphism (Metric Spaces)|homeomorphism]].
Then $f^{-1}: M_2 \to M_1$ is also a [[Definition:Homeomorphism (Metric Spaces)|homeomorphism]]. | By definition, a [[Definition:Homeomorphism (Metric Spaces)|homeomorphism]] is a [[Definition:Bijection|bijection]] such that both $f$ and $f^{-1}$ are [[Definition:Continuous on Metric Space|continuous]].
As $f$ is a [[Definition:Bijection|bijection]] then by [[Bijection iff Inverse is Bijection]], so is $f^{-1}$.
S... | Inverse of Homeomorphism between Metric Spaces is Homeomorphism | https://proofwiki.org/wiki/Inverse_of_Homeomorphism_between_Metric_Spaces_is_Homeomorphism | https://proofwiki.org/wiki/Inverse_of_Homeomorphism_between_Metric_Spaces_is_Homeomorphism | [
"Homeomorphisms (Metric Spaces)"
] | [
"Definition:Metric Space",
"Definition:Homeomorphism/Metric Spaces",
"Definition:Homeomorphism/Metric Spaces"
] | [
"Definition:Homeomorphism/Metric Spaces",
"Definition:Bijection",
"Definition:Continuous Mapping (Metric Space)/Space",
"Definition:Bijection",
"Inverse of Bijection is Bijection",
"Definition:Bijection",
"Definition:Continuous Mapping (Metric Space)/Space",
"Inverse of Inverse of Bijection",
"Categ... |
proofwiki-10291 | Composite of Homeomorphisms between Metric Spaces is Homeomorphism | Let $M_1, M_2, M_3$ be metric spaces.
Let $f: M_1 \to M_2$ and $g: M_2 \to M_3$ be homeomorphisms.
Then $g \circ f: M_1 \to M_3$ is also a homeomorphism. | By definition of homeomorphism, $f$ and $g$ are both bijections.
From Composite of Bijections is Bijection it follows that $g \circ f$ is also a bijection.
Similarly, also by definition of homeomorphism, $f$ and $g$ are both continuous mappings.
From Composite of Continuous Mappings on Metric Spaces is Continuous it fo... | Let $M_1, M_2, M_3$ be [[Definition:Metric Space|metric spaces]].
Let $f: M_1 \to M_2$ and $g: M_2 \to M_3$ be [[Definition:Homeomorphism (Metric Spaces)|homeomorphisms]].
Then $g \circ f: M_1 \to M_3$ is also a [[Definition:Homeomorphism (Metric Spaces)|homeomorphism]]. | By definition of [[Definition:Homeomorphism (Metric Spaces)|homeomorphism]], $f$ and $g$ are both [[Definition:Bijection|bijections]].
From [[Composite of Bijections is Bijection]] it follows that $g \circ f$ is also a [[Definition:Bijection|bijection]].
Similarly, also by definition of [[Definition:Homeomorphism (M... | Composite of Homeomorphisms between Metric Spaces is Homeomorphism | https://proofwiki.org/wiki/Composite_of_Homeomorphisms_between_Metric_Spaces_is_Homeomorphism | https://proofwiki.org/wiki/Composite_of_Homeomorphisms_between_Metric_Spaces_is_Homeomorphism | [
"Homeomorphisms (Metric Spaces)"
] | [
"Definition:Metric Space",
"Definition:Homeomorphism/Metric Spaces",
"Definition:Homeomorphism/Metric Spaces"
] | [
"Definition:Homeomorphism/Metric Spaces",
"Definition:Bijection",
"Composite of Bijections is Bijection",
"Definition:Bijection",
"Definition:Homeomorphism/Metric Spaces",
"Definition:Continuous Mapping (Metric Space)/Space",
"Composite of Continuous Mappings is Continuous/Corollary",
"Definition:Cont... |
proofwiki-10292 | Metrizable Space is Hausdorff | Let $T$ be a metrizable topological space.
Then $T$ is a $T_2$ (Hausdorff) space. | By definition, $T$ is homeomorphic to a topological space $\struct {S, \tau_d}$ such that $\tau_d$ is the topology induced by a metric $d$ on $S$.
From Metric Space is Hausdorff, $\struct {S, d}$ is a $T_2$ (Hausdorff) space.
As the open sets of $\struct {S, d}$ are the same as the open sets of $\struct {S, \tau_d}$, i... | Let $T$ be a [[Definition:Metrizable Space|metrizable]] [[Definition:Topological Space|topological space]].
Then $T$ is a [[Definition:Hausdorff Space|$T_2$ (Hausdorff) space]]. | By definition, $T$ is [[Definition:Homeomorphic Topological Spaces|homeomorphic]] to a [[Definition:Topological Space|topological space]] $\struct {S, \tau_d}$ such that $\tau_d$ is the [[Definition:Topology Induced by Metric|topology induced]] by a [[Definition:Metric|metric]] $d$ on $S$.
From [[Metric Space is Hausd... | Metrizable Space is Hausdorff | https://proofwiki.org/wiki/Metrizable_Space_is_Hausdorff | https://proofwiki.org/wiki/Metrizable_Space_is_Hausdorff | [
"Hausdorff Spaces",
"Metrizable Spaces"
] | [
"Definition:Metrizable Space",
"Definition:Topological Space",
"Definition:T2 Space"
] | [
"Definition:Homeomorphism/Topological Spaces",
"Definition:Topological Space",
"Definition:Topology Induced by Metric",
"Definition:Metric Space/Metric",
"Metric Space is T2",
"Definition:T2 Space",
"Definition:Open Set/Metric Space",
"Definition:Open Set/Topology",
"Definition:T2 Space",
"T2 Prop... |
proofwiki-10293 | Neighborhood of Point in Metrizable Space contains Closed Neighborhood | Let $T = \struct {S, \tau}$ be a metrizable topological space.
Let $x \in S$ be an arbitrary point of $T$.
Let $N$ be a neighborhood of $x$.
Then $N$ has as a subset a neighborhood $V$ of $x$ such that $V$ is closed. | Since $N$ is a neighborhood of $x$, there exists an open set $U \subseteq N$ containing $x$ by definition.
As $\struct {S, \tau}$ is metrizable, the set $U$ is open with respect to some metric space $\struct {S, d}$.
Hence:
:$\exists \epsilon > 0: \map {\BB_\epsilon} x \subseteq U$
where $\map {\BB_\epsilon} x$ denotes... | Let $T = \struct {S, \tau}$ be a [[Definition:Metrizable Space|metrizable]] [[Definition:Topological Space|topological space]].
Let $x \in S$ be an arbitrary point of $T$.
Let $N$ be a [[Definition:Neighborhood of Point|neighborhood]] of $x$.
Then $N$ has as a [[Definition:Subset|subset]] a [[Definition:Neighborhoo... | Since $N$ is a [[Definition:Neighborhood of Point|neighborhood]] of $x$, there exists an [[Definition:Open Set (Topology)|open]] set $U \subseteq N$ containing $x$ by definition.
As $\struct {S, \tau}$ is [[Definition:Metrizable Space|metrizable]], the set $U$ is [[Definition:Open Set (Metric Space)|open]] with respec... | Neighborhood of Point in Metrizable Space contains Closed Neighborhood | https://proofwiki.org/wiki/Neighborhood_of_Point_in_Metrizable_Space_contains_Closed_Neighborhood | https://proofwiki.org/wiki/Neighborhood_of_Point_in_Metrizable_Space_contains_Closed_Neighborhood | [
"Metrizable Spaces",
"Neighborhoods"
] | [
"Definition:Metrizable Space",
"Definition:Topological Space",
"Definition:Neighborhood (Topology)/Point",
"Definition:Subset",
"Definition:Neighborhood (Topology)/Point",
"Definition:Closed Set/Topology"
] | [
"Definition:Neighborhood (Topology)/Point",
"Definition:Open Set/Topology",
"Definition:Metrizable Space",
"Definition:Open Set/Metric Space",
"Definition:Metric Space",
"Definition:Open Ball",
"Definition:Closed Ball",
"Definition:Subset",
"Definition:Closed Ball",
"Definition:Closed Neighborhood... |
proofwiki-10294 | Underlying Set of Topological Space is Closed | Let $T = \struct {S, \tau}$ be a topological space.
Then the underlying set $S$ of $T$ is closed in $T$. | From the definition of closed set, $U$ is open in $T = \struct {S, \tau}$ {{iff}} $S \setminus U$ is closed in $T$.
From Empty Set is Element of Topology, $\O$ is open in $T$.
From Set Difference with Empty Set is Self:
:$S \setminus \O = S$
Hence $S$ is closed in $T$.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Then the [[Definition:Underlying Set of Topological Space|underlying set]] $S$ of $T$ is [[Definition:Closed Set (Topology)|closed]] in $T$. | From the definition of [[Definition:Closed Set (Topology)|closed set]], $U$ is [[Definition:Open Set (Topology)|open]] in $T = \struct {S, \tau}$ {{iff}} $S \setminus U$ is [[Definition:Closed Set (Topology)|closed]] in $T$.
From [[Empty Set is Element of Topology]], $\O$ is [[Definition:Open Set (Topology)|open]] in ... | Underlying Set of Topological Space is Closed | https://proofwiki.org/wiki/Underlying_Set_of_Topological_Space_is_Closed | https://proofwiki.org/wiki/Underlying_Set_of_Topological_Space_is_Closed | [
"Closed Sets"
] | [
"Definition:Topological Space",
"Definition:Underlying Set/Topological Space",
"Definition:Closed Set/Topology"
] | [
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology",
"Definition:Closed Set/Topology",
"Empty Set is Element of Topology",
"Definition:Open Set/Topology",
"Set Difference with Empty Set is Self",
"Definition:Closed Set/Topology"
] |
proofwiki-10295 | Point in Topological Space has Neighborhood | Let $T = \struct {S, \tau}$ be a topological space.
Let $x \in S$.
Then there exists in $T$ at least one neighborhood of $x$. | Let $x \in S$.
Then $S$ itself is a neighborhood of $x$.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $x \in S$.
Then there exists in $T$ at least one [[Definition:Neighborhood of Point|neighborhood]] of $x$. | Let $x \in S$.
Then $S$ itself is a [[Definition:Neighborhood of Point|neighborhood]] of $x$.
{{qed}} | Point in Topological Space has Neighborhood | https://proofwiki.org/wiki/Point_in_Topological_Space_has_Neighborhood | https://proofwiki.org/wiki/Point_in_Topological_Space_has_Neighborhood | [
"Neighborhoods"
] | [
"Definition:Topological Space",
"Definition:Neighborhood (Topology)/Point"
] | [
"Definition:Neighborhood (Topology)/Point"
] |
proofwiki-10296 | Point in Topological Space is Element of its Neighborhood | Let $T = \struct {S, \tau}$ be a topological space.
Let $x \in S$.
Let $N$ be a neighborhood of $x$ in $T$.
Then $a \in N$. | Trivially follows by definition of neighborhood of $a$.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $x \in S$.
Let $N$ be a [[Definition:Neighborhood of Point|neighborhood]] of $x$ in $T$.
Then $a \in N$. | Trivially follows by definition of [[Definition:Neighborhood of Point|neighborhood]] of $a$.
{{qed}} | Point in Topological Space is Element of its Neighborhood | https://proofwiki.org/wiki/Point_in_Topological_Space_is_Element_of_its_Neighborhood | https://proofwiki.org/wiki/Point_in_Topological_Space_is_Element_of_its_Neighborhood | [
"Neighborhoods"
] | [
"Definition:Topological Space",
"Definition:Neighborhood (Topology)/Point"
] | [
"Definition:Neighborhood (Topology)/Point"
] |
proofwiki-10297 | Superset of Neighborhood in Topological Space is Neighborhood | Let $T = \struct {S, \tau}$ be a topological space.
Let $x \in S$.
Let $N$ be a neighborhood of $x$ in $T$.
Let $N \subseteq N' \subseteq S$.
Then $N'$ is a neighborhood of $x$ in $T$. | By definition of neighborhood:
:$\exists U \in \tau: x \in U \subseteq N \subseteq S$
where $U$ is an open set of $T$.
By Subset Relation is Transitive:
:$U \subseteq N'$
The result follows by definition of neighborhood of $x$.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $x \in S$.
Let $N$ be a [[Definition:Neighborhood of Point|neighborhood]] of $x$ in $T$.
Let $N \subseteq N' \subseteq S$.
Then $N'$ is a [[Definition:Neighborhood of Point|neighborhood]] of $x$ in $T$. | By definition of [[Definition:Neighborhood of Point|neighborhood]]:
:$\exists U \in \tau: x \in U \subseteq N \subseteq S$
where $U$ is an [[Definition:Open Set (Topology)|open set]] of $T$.
By [[Subset Relation is Transitive]]:
:$U \subseteq N'$
The result follows by definition of [[Definition:Neighborhood of Point|... | Superset of Neighborhood in Topological Space is Neighborhood | https://proofwiki.org/wiki/Superset_of_Neighborhood_in_Topological_Space_is_Neighborhood | https://proofwiki.org/wiki/Superset_of_Neighborhood_in_Topological_Space_is_Neighborhood | [
"Neighborhoods"
] | [
"Definition:Topological Space",
"Definition:Neighborhood (Topology)/Point",
"Definition:Neighborhood (Topology)/Point"
] | [
"Definition:Neighborhood (Topology)/Point",
"Definition:Open Set/Topology",
"Subset Relation is Transitive",
"Definition:Neighborhood (Topology)/Point"
] |
proofwiki-10298 | Intersection of Neighborhoods in Topological Space is Neighborhood | Let $T = \struct {S, \tau}$ be a topological space.
Let $x \in S$.
Let $M, N$ be a neighborhoods of $x$ in $T$.
Then $M \cap N$ is a neighborhood of $x$ in $T$. | By definition of neighborhood:
:$\exists U_1 \in \tau: x \in U_1 \subseteq M$
where $U_1$ is an open set of $T$.
:$\exists U_2 \in \tau: x \in U_2 \subseteq N$
where $U_2$ is an open set of $T$.
Thus by Set Intersection Preserves Subsets:
:$U \subseteq M \cap N$
where $U = U_1 \cap U_2$
The result follows by definition... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $x \in S$.
Let $M, N$ be a [[Definition:Neighborhood of Point|neighborhoods]] of $x$ in $T$.
Then $M \cap N$ is a [[Definition:Neighborhood of Point|neighborhood]] of $x$ in $T$. | By definition of [[Definition:Neighborhood of Point|neighborhood]]:
:$\exists U_1 \in \tau: x \in U_1 \subseteq M$
where $U_1$ is an [[Definition:Open Set (Topology)|open set]] of $T$.
:$\exists U_2 \in \tau: x \in U_2 \subseteq N$
where $U_2$ is an [[Definition:Open Set (Topology)|open set]] of $T$.
Thus by [[Set I... | Intersection of Neighborhoods in Topological Space is Neighborhood | https://proofwiki.org/wiki/Intersection_of_Neighborhoods_in_Topological_Space_is_Neighborhood | https://proofwiki.org/wiki/Intersection_of_Neighborhoods_in_Topological_Space_is_Neighborhood | [
"Neighborhoods"
] | [
"Definition:Topological Space",
"Definition:Neighborhood (Topology)/Point",
"Definition:Neighborhood (Topology)/Point"
] | [
"Definition:Neighborhood (Topology)/Point",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Set Intersection Preserves Subsets",
"Definition:Neighborhood (Topology)/Point"
] |
proofwiki-10299 | Neighborhood in Topological Space has Subset Neighborhood | Let $T = \struct {S, \tau}$ be a topological space.
Let $x \in S$.
Let $N$ be a neighborhood of $x$ in $T$.
Then there exists a neighborhood $N'$ of $x$ such that:
:$(1): \quad N' \subseteq N$
:$(2): \quad N'$ is a neighborhood of each of its points. | By definition of neighborhood:
:$\exists U \in \tau: x \in U \subseteq N \subseteq S$
where $U$ is an open set of $T$.
By Set is Open iff Neighborhood of all its Points, $N' = U$ fulfils the conditions of the statement.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $x \in S$.
Let $N$ be a [[Definition:Neighborhood of Point|neighborhood]] of $x$ in $T$.
Then there exists a [[Definition:Neighborhood of Point|neighborhood]] $N'$ of $x$ such that:
:$(1): \quad N' \subseteq N$
:$(2): \quad N'$... | By definition of [[Definition:Neighborhood of Point|neighborhood]]:
:$\exists U \in \tau: x \in U \subseteq N \subseteq S$
where $U$ is an [[Definition:Open Set (Topology)|open set]] of $T$.
By [[Set is Open iff Neighborhood of all its Points]], $N' = U$ fulfils the conditions of the statement.
{{qed}} | Neighborhood in Topological Space has Subset Neighborhood | https://proofwiki.org/wiki/Neighborhood_in_Topological_Space_has_Subset_Neighborhood | https://proofwiki.org/wiki/Neighborhood_in_Topological_Space_has_Subset_Neighborhood | [
"Neighborhoods"
] | [
"Definition:Topological Space",
"Definition:Neighborhood (Topology)/Point",
"Definition:Neighborhood (Topology)/Point",
"Definition:Neighborhood (Topology)/Point"
] | [
"Definition:Neighborhood (Topology)/Point",
"Definition:Open Set/Topology",
"Set is Open iff Neighborhood of all its Points"
] |
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