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-10100 | Subset equals Preimage of Image implies Injection | Let $f: S \to T$ be a mapping.
Let $f^\to: \powerset S \to \powerset T$ be the direct image mapping of $f$.
Similarly, let $f^\gets: \powerset T \to \powerset S$ be the inverse image mapping of $f$.
Let:
:$\forall A \in \powerset S: A = \map {\paren {f^\gets \circ f^\to} } A$
Then $f$ is an injection. | Suppose that $f$ is not an injection.
Then two elements of $S$ map to the same one element of $T$.
That is:
:$\exists a_1, a_2 \in S, b \in T: \map f {a_1} = \map f {a_2} = b$
Let $A = \set {a_1}$.
Then:
{{begin-eqn}}
{{eqn | l = \map {f^\to} A
| r = \set b
| c =
}}
{{eqn | ll= \leadsto
| l = \map {f... | Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Let $f^\to: \powerset S \to \powerset T$ be the [[Definition:Direct Image Mapping of Mapping|direct image mapping]] of $f$.
Similarly, let $f^\gets: \powerset T \to \powerset S$ be the [[Definition:Inverse Image Mapping of Mapping|inverse image mapping]] of $f$.
... | Suppose that $f$ is not an [[Definition:Injection|injection]].
Then two [[Definition:Element|elements]] of $S$ map to the same one [[Definition:Element|element]] of $T$.
That is:
:$\exists a_1, a_2 \in S, b \in T: \map f {a_1} = \map f {a_2} = b$
Let $A = \set {a_1}$.
Then:
{{begin-eqn}}
{{eqn | l = \map {f^\to} A... | Subset equals Preimage of Image implies Injection/Proof 2 | https://proofwiki.org/wiki/Subset_equals_Preimage_of_Image_implies_Injection | https://proofwiki.org/wiki/Subset_equals_Preimage_of_Image_implies_Injection/Proof_2 | [
"Injections",
"Subsets",
"Subset equals Preimage of Image implies Injection"
] | [
"Definition:Mapping",
"Definition:Direct Image Mapping/Mapping",
"Definition:Inverse Image Mapping/Mapping",
"Definition:Injection"
] | [
"Definition:Injection",
"Definition:Element",
"Definition:Element",
"Rule of Transposition",
"Definition:Injection"
] |
proofwiki-10101 | Preimage of Image of Subset under Injection equals Subset | Let $f: S \to T$ be an injection.
Then:
:$\forall A \subseteq S: A = \paren {f^{-1} \circ f} \sqbrk A$
where:
:$f \sqbrk A$ denotes the image of $A$ under $f$
:$f^{-1}$ denotes the inverse of $f$
:$f^{-1} \circ f$ denotes composition of $f^{-1}$ and $f$. | Let $f$ be an injection.
From Subset of Domain is Subset of Preimage of Image, we have that:
:$\forall A \subseteq S: A \subseteq \paren {f^{-1} \circ f} \sqbrk A$
by dint of $f$ being a relation.
So what we need to do is show that:
:$\forall A \subseteq S: \paren {f^{-1} \circ f} \sqbrk A \subseteq A$
Take any $A \sub... | Let $f: S \to T$ be an [[Definition:Injection|injection]].
Then:
:$\forall A \subseteq S: A = \paren {f^{-1} \circ f} \sqbrk A$
where:
:$f \sqbrk A$ denotes the [[Definition:Image of Subset under Mapping|image of $A$ under $f$]]
:$f^{-1}$ denotes the [[Definition:Inverse of Mapping|inverse]] of $f$
:$f^{-1} \circ f... | Let $f$ be an [[Definition:Injection|injection]].
From [[Subset of Domain is Subset of Preimage of Image]], we have that:
:$\forall A \subseteq S: A \subseteq \paren {f^{-1} \circ f} \sqbrk A$
by dint of $f$ being a [[Definition:Relation|relation]].
So what we need to do is show that:
:$\forall A \subseteq S: \paren ... | Preimage of Image of Subset under Injection equals Subset | https://proofwiki.org/wiki/Preimage_of_Image_of_Subset_under_Injection_equals_Subset | https://proofwiki.org/wiki/Preimage_of_Image_of_Subset_under_Injection_equals_Subset | [
"Injections",
"Composite Mappings",
"Preimages under Mappings"
] | [
"Definition:Injection",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Inverse of Mapping",
"Definition:Composition of Mappings"
] | [
"Definition:Injection",
"Subset of Domain is Subset of Preimage of Image",
"Definition:Relation"
] |
proofwiki-10102 | Subset equals Image of Preimage implies Surjection | Let $f: S \to T$ be a mapping.
Let:
:$\forall B \subseteq T: B = \paren {f \circ f^{-1} } \sqbrk B$
where $f \sqbrk B$ denotes the image of $B$ under $f$.
Then $f$ is a surjection. | Let $f$ be such that:
:$\forall B \subseteq T: B = \paren {f \circ f^{-1} } \sqbrk B$
In particular, it holds for $T$ itself.
Hence:
{{begin-eqn}}
{{eqn | l = T
| r = \paren {f \circ f^{-1} } \sqbrk T
| c =
}}
{{eqn | l = T
| r = f \sqbrk {f^{-1} \sqbrk T}
| c = {{Defof|Composition of Mappings}... | Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Let:
:$\forall B \subseteq T: B = \paren {f \circ f^{-1} } \sqbrk B$
where $f \sqbrk B$ denotes the [[Definition:Image of Subset under Mapping|image of $B$ under $f$]].
Then $f$ is a [[Definition:Surjection|surjection]]. | Let $f$ be such that:
:$\forall B \subseteq T: B = \paren {f \circ f^{-1} } \sqbrk B$
In particular, it holds for $T$ itself.
Hence:
{{begin-eqn}}
{{eqn | l = T
| r = \paren {f \circ f^{-1} } \sqbrk T
| c =
}}
{{eqn | l = T
| r = f \sqbrk {f^{-1} \sqbrk T}
| c = {{Defof|Composition of Mappin... | Subset equals Image of Preimage implies Surjection/Proof 1 | https://proofwiki.org/wiki/Subset_equals_Image_of_Preimage_implies_Surjection | https://proofwiki.org/wiki/Subset_equals_Image_of_Preimage_implies_Surjection/Proof_1 | [
"Surjections",
"Subset equals Image of Preimage implies Surjection"
] | [
"Definition:Mapping",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Surjection"
] | [
"Image of Subset under Mapping is Subset of Image",
"Definition:Set Equality/Definition 2",
"Definition:Surjection"
] |
proofwiki-10103 | Subset equals Image of Preimage implies Surjection | Let $f: S \to T$ be a mapping.
Let:
:$\forall B \subseteq T: B = \paren {f \circ f^{-1} } \sqbrk B$
where $f \sqbrk B$ denotes the image of $B$ under $f$.
Then $f$ is a surjection. | Suppose $f$ is not a surjection.
$T$ must have at least two elements for this to be the case.
Let one of these two elements not be the image of any element of $S$.
That is, let $b_1, b_2 \in T$ such that:
:$\exists a \in S: f \paren a = b_1$
:$\nexists x \in S: f \paren x = b_2$
Let $B = \set {b_1, b_2}$.
Then:
{{begin... | Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Let:
:$\forall B \subseteq T: B = \paren {f \circ f^{-1} } \sqbrk B$
where $f \sqbrk B$ denotes the [[Definition:Image of Subset under Mapping|image of $B$ under $f$]].
Then $f$ is a [[Definition:Surjection|surjection]]. | Suppose $f$ is not a [[Definition:Surjection|surjection]].
$T$ must have at least two [[Definition:Element|elements]] for this to be the case.
Let one of these two [[Definition:Element|elements]] not be the [[Definition:Image of Element under Mapping|image]] of any [[Definition:Element|element]] of $S$.
That is, let... | Subset equals Image of Preimage implies Surjection/Proof 2 | https://proofwiki.org/wiki/Subset_equals_Image_of_Preimage_implies_Surjection | https://proofwiki.org/wiki/Subset_equals_Image_of_Preimage_implies_Surjection/Proof_2 | [
"Surjections",
"Subset equals Image of Preimage implies Surjection"
] | [
"Definition:Mapping",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Surjection"
] | [
"Definition:Surjection",
"Definition:Element",
"Definition:Element",
"Definition:Image (Set Theory)/Mapping/Element",
"Definition:Element",
"Definition:Preimage/Mapping/Element",
"Rule of Transposition",
"Definition:Surjection"
] |
proofwiki-10104 | Image of Preimage of Subset under Surjection equals Subset | Let $f: S \to T$ be a surjection.
Then:
:$\forall B \subseteq T: B = \paren {f \circ f^{-1} } \sqbrk B$
where:
:$f \sqbrk B$ denotes the image of $B$ under $f$
:$f^{-1}$ denotes the inverse of $f$
:$f \circ f^{-1}$ denotes composition of $f$ and $f^{-1}$. | Let $g$ be a surjection.
Let $B \subseteq T$.
Let $b \in B$.
Then:
{{begin-eqn}}
{{eqn | q = \exists a \in S
| l = b
| r = \map f a
| c = {{Defof|Surjection}}
}}
{{eqn | ll= \leadsto
| l = a
| o = \in
| r = f^{-1} \sqbrk B
| c = {{Defof|Preimage of Subset under Mapping}}
}}
{{... | Let $f: S \to T$ be a [[Definition:Surjection|surjection]].
Then:
:$\forall B \subseteq T: B = \paren {f \circ f^{-1} } \sqbrk B$
where:
:$f \sqbrk B$ denotes the [[Definition:Image of Subset under Mapping|image of $B$ under $f$]]
:$f^{-1}$ denotes the [[Definition:Inverse of Mapping|inverse]] of $f$
:$f \circ f^{-1... | Let $g$ be a [[Definition:Surjection|surjection]].
Let $B \subseteq T$.
Let $b \in B$.
Then:
{{begin-eqn}}
{{eqn | q = \exists a \in S
| l = b
| r = \map f a
| c = {{Defof|Surjection}}
}}
{{eqn | ll= \leadsto
| l = a
| o = \in
| r = f^{-1} \sqbrk B
| c = {{Defof|Preimage o... | Image of Preimage of Subset under Surjection equals Subset | https://proofwiki.org/wiki/Image_of_Preimage_of_Subset_under_Surjection_equals_Subset | https://proofwiki.org/wiki/Image_of_Preimage_of_Subset_under_Surjection_equals_Subset | [
"Surjections",
"Preimages under Mappings",
"Composite Mappings"
] | [
"Definition:Surjection",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Inverse of Mapping",
"Definition:Composition of Mappings"
] | [
"Definition:Surjection",
"Subset of Codomain is Superset of Image of Preimage",
"Definition:Set Equality/Definition 2"
] |
proofwiki-10105 | Intersection of Image with Subset of Codomain | Let $S$ and $T$ be sets.
Let $f: S \to T$ be a mapping.
Let $A \subseteq S$ and $B \subseteq T$.
Then:
:$f \sqbrk {A \cap f^{-1} \sqbrk B} = f \sqbrk A \cap B$ | {{begin-eqn}}
{{eqn | l = f \sqbrk {A \cap f^{-1} \sqbrk B}
| r = \set {\map f x: x \in A \cap f^{-1} \sqbrk B}
| c = {{Defof|Image of Subset under Mapping}}
}}
{{eqn | r = \set {\map f x: x \in A \land x \in f^{-1} \sqbrk B}
| c = {{Defof|Set Intersection}}
}}
{{eqn | r = \set {\map f x: x \in A \lan... | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Let $A \subseteq S$ and $B \subseteq T$.
Then:
:$f \sqbrk {A \cap f^{-1} \sqbrk B} = f \sqbrk A \cap B$ | {{begin-eqn}}
{{eqn | l = f \sqbrk {A \cap f^{-1} \sqbrk B}
| r = \set {\map f x: x \in A \cap f^{-1} \sqbrk B}
| c = {{Defof|Image of Subset under Mapping}}
}}
{{eqn | r = \set {\map f x: x \in A \land x \in f^{-1} \sqbrk B}
| c = {{Defof|Set Intersection}}
}}
{{eqn | r = \set {\map f x: x \in A \lan... | Intersection of Image with Subset of Codomain | https://proofwiki.org/wiki/Intersection_of_Image_with_Subset_of_Codomain | https://proofwiki.org/wiki/Intersection_of_Image_with_Subset_of_Codomain | [
"Images",
"Set Intersection"
] | [
"Definition:Set",
"Definition:Mapping"
] | [] |
proofwiki-10106 | Projection is Injection iff Factor is Singleton | Let $S_1, S_2, \ldots, S_n$ be non-empty sets.
Let $\ds S = \prod_{i \mathop = 1}^n S_i$ be the cartesian product of $S_1, S_2, \ldots, S_n$.
Let $\pr_j: S \to S_j$ be the $j$th projection on $S$.
Then $\pr_j$ is an injection {{iff}} $S_k$ is a singleton for all $k \in \set {1, 2, \dotsc, n}$ where $k \ne j$. | === Sufficient Condition ===
Suppose $S_k = \set {s_k}$ for all $k \in \set {1, 2, \dotsc, n}$ where $k \ne j$.
Let $\map {\pr_j} x = \map {\pr_j} y = z$ for $x, y \in S$.
Then by definition of $j$th projection, $x, y \in S$ are given by:
:$x = \tuple {s_1, s_2, \dotsc, s_{j - 1}, z, s_{j + 1}, \dotsc, s_n}$
:$y = \tup... | Let $S_1, S_2, \ldots, S_n$ be [[Definition:Non-Empty Set|non-empty]] [[Definition:Set|sets]].
Let $\ds S = \prod_{i \mathop = 1}^n S_i$ be the [[Definition:Finite Cartesian Product|cartesian product]] of $S_1, S_2, \ldots, S_n$.
Let $\pr_j: S \to S_j$ be the [[Definition:Projection (Mapping Theory)|$j$th projection]... | === Sufficient Condition ===
Suppose $S_k = \set {s_k}$ for all $k \in \set {1, 2, \dotsc, n}$ where $k \ne j$.
Let $\map {\pr_j} x = \map {\pr_j} y = z$ for $x, y \in S$.
Then by definition of [[Definition:Projection (Mapping Theory)|$j$th projection]], $x, y \in S$ are given by:
:$x = \tuple {s_1, s_2, \dotsc, s_{... | Projection is Injection iff Factor is Singleton | https://proofwiki.org/wiki/Projection_is_Injection_iff_Factor_is_Singleton | https://proofwiki.org/wiki/Projection_is_Injection_iff_Factor_is_Singleton | [
"Projections",
"Injections",
"Singletons",
"Projection is Injection iff Factor is Singleton"
] | [
"Definition:Non-Empty Set",
"Definition:Set",
"Definition:Cartesian Product/Finite",
"Definition:Projection (Mapping Theory)",
"Definition:Injection",
"Definition:Singleton"
] | [
"Definition:Projection (Mapping Theory)",
"Definition:Injection",
"Definition:Injection"
] |
proofwiki-10107 | Preimage of Element under Projection | Let $A$ and $B$ be sets.
Let $A \times B$ be the cartesian product of $A$ and $B$.
Let $\pr_1: A \times B \to A$ be the first projection of $A \times B$.
Let $a \in A$.
Then:
:$\pr_1^{-1} \sqbrk {\set a} = \set {\tuple {a, b}: b \in B}$
that is:
:$\pr_1^{-1} \sqbrk {\set a} = \set a \times B$ | Directly apparent from the definition of cartesian product.
{{qed}} | Let $A$ and $B$ be [[Definition:Set|sets]].
Let $A \times B$ be the [[Definition:Cartesian Product|cartesian product]] of $A$ and $B$.
Let $\pr_1: A \times B \to A$ be the [[Definition:First Projection|first projection]] of $A \times B$.
Let $a \in A$.
Then:
:$\pr_1^{-1} \sqbrk {\set a} = \set {\tuple {a, b}: b \i... | Directly apparent from the definition of [[Definition:Cartesian Product|cartesian product]].
{{qed}} | Preimage of Element under Projection | https://proofwiki.org/wiki/Preimage_of_Element_under_Projection | https://proofwiki.org/wiki/Preimage_of_Element_under_Projection | [
"Projections"
] | [
"Definition:Set",
"Definition:Cartesian Product",
"Definition:Projection (Mapping Theory)/First Projection"
] | [
"Definition:Cartesian Product"
] |
proofwiki-10108 | Divisors of Factorial | Let $n \in \N_{>0}$.
Then all natural numbers less than or equal to $n$ are divisors of $n!$:
:$\forall k \in \left\{{1, 2, \ldots, n}\right\}: n! \equiv 0 \pmod k$ | From the definition of factorial:
:$n! = 1 \times 2 \times \cdots \times \left({n-1}\right) \times n$
Thus every number less than $n$ appears as a divisor of $n!$.
The result follows from definition of congruence.
{{qed}}
Category:Factorials
Category:Divisors
3p1tjojrrsp4gejtpfuz94uxurucs8o | Let $n \in \N_{>0}$.
Then all [[Definition:Natural Numbers|natural numbers]] less than or equal to $n$ are [[Definition:Divisor of Integer|divisors]] of $n!$:
:$\forall k \in \left\{{1, 2, \ldots, n}\right\}: n! \equiv 0 \pmod k$ | From the definition of [[Definition:Factorial|factorial]]:
:$n! = 1 \times 2 \times \cdots \times \left({n-1}\right) \times n$
Thus every number less than $n$ appears as a [[Definition:Divisor of Integer|divisor]] of $n!$.
The result follows from definition of [[Definition:Congruence (Number Theory)/Integers/Integer... | Divisors of Factorial | https://proofwiki.org/wiki/Divisors_of_Factorial | https://proofwiki.org/wiki/Divisors_of_Factorial | [
"Factorials",
"Divisors"
] | [
"Definition:Natural Numbers",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Factorial",
"Definition:Divisor (Algebra)/Integer",
"Definition:Congruence (Number Theory)/Integers/Integer Multiple",
"Category:Factorials",
"Category:Divisors"
] |
proofwiki-10109 | Positive Difference Relation on Reals is Transitive | Let $P \subseteq \R$ be a subset of the real numbers such that:
{{begin-itemize}}
{{item|(1):|$1 \in P$}}
{{item|(2):|$a, b \in P \implies a + b \in P$}}
{{item|(3):|For all $x \in \R$, exactly one of these is true:
{{begin-itemize}}
{{item||$x \in P$}}
{{item||$x {{=}} 0$}}
{{item||$-x \in P$}}
{{end-itemize}} }}
{{en... | Let $a - b \in P$ and $b - c \in P$.
By condition $(2)$:
:$\paren {a - b} + \paren {b - c} \in P$
Simplifying:
:$a - c \in P$
The result follows by definition of transitive relation.
{{qed}} | Let $P \subseteq \R$ be a [[Definition:Subset|subset]] of the [[Definition:Real Numbers|real numbers]] such that:
{{begin-itemize}}
{{item|(1):|$1 \in P$}}
{{item|(2):|$a, b \in P \implies a + b \in P$}}
{{item|(3):|For all $x \in \R$, exactly one of these is true:
{{begin-itemize}}
{{item||$x \in P$}}
{{item||$x {{=}}... | Let $a - b \in P$ and $b - c \in P$.
By condition $(2)$:
:$\paren {a - b} + \paren {b - c} \in P$
Simplifying:
:$a - c \in P$
The result follows by definition of [[Definition:Transitive Relation|transitive relation]].
{{qed}} | Positive Difference Relation on Reals is Transitive | https://proofwiki.org/wiki/Positive_Difference_Relation_on_Reals_is_Transitive | https://proofwiki.org/wiki/Positive_Difference_Relation_on_Reals_is_Transitive | [
"Transitive Relations"
] | [
"Definition:Subset",
"Definition:Real Number",
"Definition:Relation",
"Definition:Transitive Relation"
] | [
"Definition:Transitive Relation"
] |
proofwiki-10110 | Repeated Composition of Injection is Injection | Let $S$ be a set.
Let $f: S \to S$ be an injection.
Let the sequence of mappings:
:$f^0, f^1, f^2, \ldots, f^n, \ldots$
be defined as:
:<nowiki>$\forall n \in \N: \map {f^n} x = \begin {cases}
x & : n = 0 \\
\map f x & : n = 1 \\
\map f {\map {f^{n - 1} } x} & : n > 1
\end{cases}$</nowiki>
Then for all $n \in \N$, $f^n... | Proof by induction:
For all $n \in \N_{\ge 0}$, let $\map P n$ be the proposition:
:$f^n$ is an injection.
$\map P 0$ is true, as this is the case Identity Mapping is Injection.
$\map P 1$ is true, as this is the assertion that $f$ is an injection. | Let $S$ be a [[Definition:Set|set]].
Let $f: S \to S$ be an [[Definition:Injection|injection]].
Let the [[Definition:Sequence|sequence]] of [[Definition:Mapping|mappings]]:
:$f^0, f^1, f^2, \ldots, f^n, \ldots$
be defined as:
:<nowiki>$\forall n \in \N: \map {f^n} x = \begin {cases}
x & : n = 0 \\
\map f x & : n = 1 ... | Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \N_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$f^n$ is an [[Definition:Injection|injection]].
$\map P 0$ is true, as this is the case [[Identity Mapping is Injection]].
$\map P 1$ is true, as this is the asserti... | Repeated Composition of Injection is Injection | https://proofwiki.org/wiki/Repeated_Composition_of_Injection_is_Injection | https://proofwiki.org/wiki/Repeated_Composition_of_Injection_is_Injection | [
"Injections",
"Composite Mappings"
] | [
"Definition:Set",
"Definition:Injection",
"Definition:Sequence",
"Definition:Mapping",
"Definition:Injection"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Injection",
"Identity Mapping is Injection",
"Definition:Injection",
"Definition:Injection",
"Definition:Injection",
"Definition:Injection",
"Definition:Injection",
"Definition:Injection",
"Definition:Injection",
"Pri... |
proofwiki-10111 | Images of Elements under Repeated Composition of Injection form Equivalence Classes | Let $S$ be a set.
Let $f: S \to S$ be an injection.
Let the sequence of mappings:
:$f^0, f^1, f^2, \ldots, f^n, \ldots$
be defined as:
:$\forall n \in \N: \map {f^n} x = \begin {cases}
x & : n = 0 \\
\map f x & : n = 1 \\
\map f {\map {f^{n - 1} } x} & : n > 1
\end {cases}$
Let $\RR \subseteq S \times S$ be the relatio... | Checking in turn each of the criteria for equivalence: | Let $S$ be a [[Definition:Set|set]].
Let $f: S \to S$ be an [[Definition:Injection|injection]].
Let the [[Definition:Sequence|sequence]] of [[Definition:Mapping|mappings]]:
:$f^0, f^1, f^2, \ldots, f^n, \ldots$
be defined as:
:$\forall n \in \N: \map {f^n} x = \begin {cases}
x & : n = 0 \\
\map f x & : n = 1 \\
\map ... | Checking in turn each of the criteria for [[Definition:Equivalence Relation|equivalence]]: | Images of Elements under Repeated Composition of Injection form Equivalence Classes | https://proofwiki.org/wiki/Images_of_Elements_under_Repeated_Composition_of_Injection_form_Equivalence_Classes | https://proofwiki.org/wiki/Images_of_Elements_under_Repeated_Composition_of_Injection_form_Equivalence_Classes | [
"Injections",
"Composite Mappings"
] | [
"Definition:Set",
"Definition:Injection",
"Definition:Sequence",
"Definition:Mapping",
"Definition:Relation",
"Definition:Equivalence Relation"
] | [
"Definition:Equivalence Relation",
"Definition:Equivalence Relation"
] |
proofwiki-10112 | Composition of Repeated Compositions of Injections | Let $S$ be a set.
Let $f: S \to S$ be an injection.
Let the sequence of mappings:
:$f^0, f^1, f^2, \ldots, f^n, \ldots$
be defined as:
:$\forall n \in \N: f^n \left({x}\right) = \begin{cases}
x & : n = 0 \\
f \left({x}\right) & : n = 1 \\
f \left({f^{n-1} \left({x}\right)}\right) & : n > 1
\end{cases}$
Then:
:$\forall ... | Proof by induction:
Let $m \in \Z_{\ge 0}$ be given.
For all $n \in \Z_{\ge 0}$, let $P \left({n}\right)$ be the proposition:
:$f^n \circ f^m = f^{m + n}$
$P \left({0}\right)$ is true, as this is the case:
{{begin-eqn}}
{{eqn | l = f^0 \circ f^m
| r = I_S \circ f^m
| c = where $I_S$ is the identity mapping
... | Let $S$ be a [[Definition:Set|set]].
Let $f: S \to S$ be an [[Definition:Injection|injection]].
Let the [[Definition:Sequence|sequence]] of [[Definition:Mapping|mappings]]:
:$f^0, f^1, f^2, \ldots, f^n, \ldots$
be defined as:
:$\forall n \in \N: f^n \left({x}\right) = \begin{cases}
x & : n = 0 \\
f \left({x}\right) &... | Proof by [[Principle of Mathematical Induction|induction]]:
Let $m \in \Z_{\ge 0}$ be given.
For all $n \in \Z_{\ge 0}$, let $P \left({n}\right)$ be the [[Definition:Proposition|proposition]]:
:$f^n \circ f^m = f^{m + n}$
$P \left({0}\right)$ is true, as this is the case:
{{begin-eqn}}
{{eqn | l = f^0 \circ f^m
... | Composition of Repeated Compositions of Injections | https://proofwiki.org/wiki/Composition_of_Repeated_Compositions_of_Injections | https://proofwiki.org/wiki/Composition_of_Repeated_Compositions_of_Injections | [
"Injections",
"Composite Mappings"
] | [
"Definition:Set",
"Definition:Injection",
"Definition:Sequence",
"Definition:Mapping",
"Definition:Composition of Mappings"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Identity Mapping",
"Identity Mapping is Left Identity",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction"
] |
proofwiki-10113 | Derivative Function on Set of Functions induces Equivalence Relation | Let $X$ be the set of real functions $f: \R \to \R$ which possess continuous derivatives.
Let $\RR \subseteq X \times X$ be the relation on $X$ defined as:
:$\RR = \set {\tuple {f, g} \in X \times X: D f = D g}$
where $D f$ denotes the first derivative of $f$.
Then $\RR$ is an equivalence relation. | Checking in turn each of the criteria for equivalence: | Let $X$ be the [[Definition:Set|set]] of [[Definition:Real Function|real functions]] $f: \R \to \R$ which [[Definition:Continuously Differentiable|possess continuous derivatives]].
Let $\RR \subseteq X \times X$ be the [[Definition:Relation|relation]] on $X$ defined as:
:$\RR = \set {\tuple {f, g} \in X \times X: D f ... | Checking in turn each of the criteria for [[Definition:Equivalence Relation|equivalence]]: | Derivative Function on Set of Functions induces Equivalence Relation | https://proofwiki.org/wiki/Derivative_Function_on_Set_of_Functions_induces_Equivalence_Relation | https://proofwiki.org/wiki/Derivative_Function_on_Set_of_Functions_induces_Equivalence_Relation | [
"Differential Calculus",
"Equivalence Relations"
] | [
"Definition:Set",
"Definition:Real Function",
"Definition:Continuously Differentiable",
"Definition:Relation",
"Definition:Derivative/Real Function/Derivative on Interval",
"Definition:Equivalence Relation"
] | [
"Definition:Equivalence Relation",
"Definition:Equivalence Relation"
] |
proofwiki-10114 | Factorial Divides Product of Successive Numbers | Let $m, n \in \N_{\ge 1}$ be natural numbers
Let $m^{\overline n}$ be $m$ to the power of $n$ rising.
Then:
:$m^{\overline n} \equiv 0 \bmod n!$
That is, the factorial of $n$ divides the product of $n$ successive numbers. | {{questionable|This is not enough. It might superficially be that e.g. the factors of $2$ and $6$ coincide. Effectively this argument reduces to the theorem; it's almost circular}}
Let $m \in \N_{\ge 1}$.
Consider the set:
:$S = \set{m, m + 1, m + 2, \ldots, m + n - 1}$
Note $S$ has $n$ elements.
By Set of Successive N... | Let $m, n \in \N_{\ge 1}$ be [[Definition:Natural Numbers|natural numbers]]
Let $m^{\overline n}$ be [[Definition:Rising Factorial|$m$ to the power of $n$ rising]].
Then:
:$m^{\overline n} \equiv 0 \bmod n!$
That is, the [[Definition:Factorial|factorial]] of $n$ [[Definition:Divisor of Integer|divides]] the product... | {{questionable|This is not enough. It might superficially be that e.g. the factors of $2$ and $6$ coincide. Effectively this argument reduces to the theorem; it's almost circular}}
Let $m \in \N_{\ge 1}$.
Consider the [[Definition:Set|set]]:
:$S = \set{m, m + 1, m + 2, \ldots, m + n - 1}$
Note $S$ has $n$ [[Definiti... | Factorial Divides Product of Successive Numbers | https://proofwiki.org/wiki/Factorial_Divides_Product_of_Successive_Numbers | https://proofwiki.org/wiki/Factorial_Divides_Product_of_Successive_Numbers | [
"Number Theory",
"Factorials"
] | [
"Definition:Natural Numbers",
"Definition:Rising Factorial",
"Definition:Factorial",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Set",
"Definition:Element",
"Set of Successive Numbers contains Unique Multiple",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Category:Number Theory",
"Category:Factorials"
] |
proofwiki-10115 | Binomial Coefficient is Integer | Let $\dbinom n k$ be a binomial coefficient.
Then $\dbinom n k$ is an integer. | If it is not the case that $0 \le k \le n$, then the result holds trivially.
So let $0 \le k \le n$.
By the definition of binomial coefficients:
{{begin-eqn}}
{{eqn | l = \binom n k
| r = \frac {n!} {k! \paren {n - k}!}
}}
{{eqn | r = \frac {n \paren {n - 1} \paren {n - 2} \cdots \paren {n - k + 1} } {k!}
}}
{{en... | Let $\dbinom n k$ be a [[Definition:Binomial Coefficient|binomial coefficient]].
Then $\dbinom n k$ is an [[Definition:Integer|integer]]. | If it is not the case that $0 \le k \le n$, then the result holds trivially.
So let $0 \le k \le n$.
By the definition of [[Definition:Binomial Coefficient|binomial coefficients]]:
{{begin-eqn}}
{{eqn | l = \binom n k
| r = \frac {n!} {k! \paren {n - k}!}
}}
{{eqn | r = \frac {n \paren {n - 1} \paren {n - 2} \... | Binomial Coefficient is Integer/Proof 1 | https://proofwiki.org/wiki/Binomial_Coefficient_is_Integer | https://proofwiki.org/wiki/Binomial_Coefficient_is_Integer/Proof_1 | [
"Binomial Coefficient is Integer",
"Binomial Coefficients"
] | [
"Definition:Binomial Coefficient",
"Definition:Integer"
] | [
"Definition:Binomial Coefficient",
"Definition:Fraction/Numerator",
"Definition:Multiplication/Integers",
"Definition:Integer",
"Factorial Divides Product of Successive Numbers",
"Definition:Divisor (Algebra)/Integer"
] |
proofwiki-10116 | Binomial Coefficient is Integer | Let $\dbinom n k$ be a binomial coefficient.
Then $\dbinom n k$ is an integer. | The result follows by Pascal's Rule and Integer Addition is Closed.
{{qed}} | Let $\dbinom n k$ be a [[Definition:Binomial Coefficient|binomial coefficient]].
Then $\dbinom n k$ is an [[Definition:Integer|integer]]. | The result follows by [[Pascal's Rule]] and [[Integer Addition is Closed]].
{{qed}} | Binomial Coefficient is Integer/Proof 2 | https://proofwiki.org/wiki/Binomial_Coefficient_is_Integer | https://proofwiki.org/wiki/Binomial_Coefficient_is_Integer/Proof_2 | [
"Binomial Coefficient is Integer",
"Binomial Coefficients"
] | [
"Definition:Binomial Coefficient",
"Definition:Integer"
] | [
"Pascal's Rule",
"Integer Addition is Closed"
] |
proofwiki-10117 | Equivalence Classes induced by Derivative Function on Set of Functions | Let $X$ be the set of real functions $f: \R \to \R$ which possess continuous derivatives.
Let $\RR \subseteq X \times X$ be the equivalence relation on $X$ defined as:
:$\RR = \set {\tuple {f, g} \in X \times X: D f = D g}$
where $D f$ denotes the first derivative of $f$.
Then the equivalence classes of $\RR$ are defin... | Follows directly from Derivative Function on Set of Functions induces Equivalence Relation.
{{qed}} | Let $X$ be the set of [[Definition:Real Function|real functions]] $f: \R \to \R$ which [[Definition:Continuously Differentiable|possess continuous derivatives]].
Let $\RR \subseteq X \times X$ be the [[Definition:Equivalence Relation|equivalence relation]] on $X$ defined as:
:$\RR = \set {\tuple {f, g} \in X \times X:... | Follows directly from [[Derivative Function on Set of Functions induces Equivalence Relation]].
{{qed}} | Equivalence Classes induced by Derivative Function on Set of Functions | https://proofwiki.org/wiki/Equivalence_Classes_induced_by_Derivative_Function_on_Set_of_Functions | https://proofwiki.org/wiki/Equivalence_Classes_induced_by_Derivative_Function_on_Set_of_Functions | [
"Differential Calculus",
"Examples of Equivalence Classes"
] | [
"Definition:Real Function",
"Definition:Continuously Differentiable",
"Definition:Equivalence Relation",
"Definition:Derivative/Real Function/Derivative on Interval",
"Definition:Equivalence Class",
"Definition:Set",
"Definition:Real Function",
"Definition:Real Number",
"Definition:Constant"
] | [
"Derivative Function on Set of Functions induces Equivalence Relation"
] |
proofwiki-10118 | Set of Mappings which map to Same Element induces Equivalence Relation | Let $X$ and $Y$ be sets.
Let $E$ be the set of all mappings from $X$ to $Y$.
Let $b \in X$.
Let $\RR \subseteq E \times E$ be the relation on $E$ defined as:
:$\RR := \set {\tuple {f, g} \in \RR: \map f b = \map g b}$
Then $\RR$ is an equivalence relation. | Checking in turn each of the criteria for equivalence: | Let $X$ and $Y$ be [[Definition:Set|sets]].
Let $E$ be the [[Definition:Set|set]] of all [[Definition:Mapping|mappings]] from $X$ to $Y$.
Let $b \in X$.
Let $\RR \subseteq E \times E$ be the [[Definition:Relation|relation]] on $E$ defined as:
:$\RR := \set {\tuple {f, g} \in \RR: \map f b = \map g b}$
Then $\RR$ i... | Checking in turn each of the criteria for [[Definition:Equivalence Relation|equivalence]]: | Set of Mappings which map to Same Element induces Equivalence Relation | https://proofwiki.org/wiki/Set_of_Mappings_which_map_to_Same_Element_induces_Equivalence_Relation | https://proofwiki.org/wiki/Set_of_Mappings_which_map_to_Same_Element_induces_Equivalence_Relation | [
"Mapping Theory",
"Equivalence Relations"
] | [
"Definition:Set",
"Definition:Set",
"Definition:Mapping",
"Definition:Relation",
"Definition:Equivalence Relation"
] | [
"Definition:Equivalence Relation",
"Definition:Equivalence Relation"
] |
proofwiki-10119 | Renaming Mapping from Set of Mappings on Single Element | Let $X$ and $Y$ be sets.
Let $E$ be the set of all mappings from $X$ to $Y$.
Let $b \in X$.
Let $\RR \subseteq E \times E$ be the relation on $E$ defined as:
:$\RR := \set {\tuple {f, g} \in \RR: \map f b = \map g b}$
Let $e_b: E / \RR \to Y$ be the renaming mapping induced by $\RR$.
Then $e_b$ is a bijection. | This is an instance of Renaming Mapping is Bijection.
{{qed}} | Let $X$ and $Y$ be [[Definition:Set|sets]].
Let $E$ be the [[Definition:Set|set]] of all [[Definition:Mapping|mappings]] from $X$ to $Y$.
Let $b \in X$.
Let $\RR \subseteq E \times E$ be the [[Definition:Relation|relation]] on $E$ defined as:
:$\RR := \set {\tuple {f, g} \in \RR: \map f b = \map g b}$
Let $e_b: E /... | This is an instance of [[Renaming Mapping is Bijection]].
{{qed}} | Renaming Mapping from Set of Mappings on Single Element | https://proofwiki.org/wiki/Renaming_Mapping_from_Set_of_Mappings_on_Single_Element | https://proofwiki.org/wiki/Renaming_Mapping_from_Set_of_Mappings_on_Single_Element | [
"Mapping Theory",
"Equivalence Relations"
] | [
"Definition:Set",
"Definition:Set",
"Definition:Mapping",
"Definition:Relation",
"Definition:Renaming Mapping",
"Definition:Bijection"
] | [
"Renaming Mapping is Bijection"
] |
proofwiki-10120 | Equivalence of Definitions of Inverse Mapping | Let $S$ and $T$ be sets.
{{TFAE|def = Inverse Mapping}} | === Definition 1 implies Definition 2 ===
Let $f^{-1}: T \to S$ be an inverse mapping of $f: S \to T$ by definition 1.
From Mapping is Injection and Surjection iff Inverse is Mapping it follows that $f^{-1}$ is a bijection.
By Composite of Bijection with Inverse is Identity Mapping:
: $f^{-1} \circ f = I_S$
: $f \circ ... | Let $S$ and $T$ be [[Definition:Set|sets]].
{{TFAE|def = Inverse Mapping}} | === Definition 1 implies Definition 2 ===
Let $f^{-1}: T \to S$ be an [[Definition:Inverse Mapping/Definition 1|inverse mapping of $f: S \to T$ by definition 1]].
From [[Mapping is Injection and Surjection iff Inverse is Mapping]] it follows that $f^{-1}$ is a [[Definition:Bijection|bijection]].
By [[Composite of Bi... | Equivalence of Definitions of Inverse Mapping | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Inverse_Mapping | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Inverse_Mapping | [
"Inverse Mappings"
] | [
"Definition:Set"
] | [
"Definition:Inverse Mapping/Definition 1",
"Mapping is Injection and Surjection iff Inverse is Mapping",
"Definition:Bijection",
"Composite of Bijection with Inverse is Identity Mapping",
"Definition:Inverse Mapping/Definition 2",
"Definition:Inverse Mapping/Definition 2",
"Definition:Bijection",
"Map... |
proofwiki-10121 | Inverse Mapping is Bijection | Let $S$ and $T$ be sets.
Let $f: S \to T$ and $g: T \to S$ be inverse mappings of each other.
Then $f$ and $g$ are bijections. | From Inverse is Mapping implies Mapping is Injection and Surjection:
:$f$ is both an injection and a surjection.
Again from Inverse is Mapping implies Mapping is Injection and Surjection:
:$g$ is both an injection and a surjection.
The result follows by definition of bijection.
{{qed}} | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $f: S \to T$ and $g: T \to S$ be [[Definition:Inverse Mapping|inverse mappings]] of each other.
Then $f$ and $g$ are [[Definition:Bijection|bijections]]. | From [[Inverse is Mapping implies Mapping is Injection and Surjection]]:
:$f$ is both an [[Definition:Injection|injection]] and a [[Definition:Surjection|surjection]].
Again from [[Inverse is Mapping implies Mapping is Injection and Surjection]]:
:$g$ is both an [[Definition:Injection|injection]] and a [[Definition... | Inverse Mapping is Bijection | https://proofwiki.org/wiki/Inverse_Mapping_is_Bijection | https://proofwiki.org/wiki/Inverse_Mapping_is_Bijection | [
"Bijections",
"Inverse Mappings"
] | [
"Definition:Set",
"Definition:Inverse Mapping",
"Definition:Bijection"
] | [
"Inverse is Mapping implies Mapping is Injection and Surjection",
"Definition:Injection",
"Definition:Surjection",
"Inverse is Mapping implies Mapping is Injection and Surjection",
"Definition:Injection",
"Definition:Surjection",
"Definition:Bijection/Definition 1"
] |
proofwiki-10122 | Mapping is Injection and Surjection iff Inverse is Mapping/Proof 2 | Let $S$ and $T$ be sets.
Let $f: S \to T$ be a mapping.
Then:
: $f: S \to T$ can be defined as a bijection in the sense that:
::$(1): \quad f$ is an injection
::$(2): \quad f$ is a surjection.
{{iff}}:
:the inverse $f^{-1}$ of $f$ is such that:
::for each $y \in T$, the preimage $\map {f^{-1} } y$ has exactly one eleme... | === Necessary Condition ===
{{:Inverse of Injective and Surjective Mapping is Mapping/Proof 2}}{{qed|lemma}}
=== Sufficient Condition ===
Let $f^{-1}: T \to S$ be a mapping.
By Inverse Mapping is Bijection, both $f$ and $f^{-1}$ are bijections.
Hence, in particular, $f$ is a bijection.
{{qed}} | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Then:
: $f: S \to T$ can be defined as a [[Definition:Bijection/Definition 1|bijection]] in the sense that:
::$(1): \quad f$ is an [[Definition:Injection|injection]]
::$(2): \quad f$ is a [[Definition:Surjection|surjecti... | === [[Inverse of Injective and Surjective Mapping is Mapping/Proof 2|Necessary Condition]] ===
{{:Inverse of Injective and Surjective Mapping is Mapping/Proof 2}}{{qed|lemma}}
=== Sufficient Condition ===
Let $f^{-1}: T \to S$ be a [[Definition:Mapping|mapping]].
By [[Inverse Mapping is Bijection]], both $f$ and $f... | Mapping is Injection and Surjection iff Inverse is Mapping/Proof 2 | https://proofwiki.org/wiki/Mapping_is_Injection_and_Surjection_iff_Inverse_is_Mapping/Proof_2 | https://proofwiki.org/wiki/Mapping_is_Injection_and_Surjection_iff_Inverse_is_Mapping/Proof_2 | [
"Mapping is Injection and Surjection iff Inverse is Mapping"
] | [
"Definition:Set",
"Definition:Mapping",
"Definition:Bijection/Definition 1",
"Definition:Injection",
"Definition:Surjection",
"Definition:Inverse of Mapping",
"Definition:Universal Quantifier",
"Definition:Preimage/Mapping/Element",
"Definition:Unique",
"Definition:Element",
"Definition:Mapping"... | [
"Inverse of Injective and Surjective Mapping is Mapping/Proof 2",
"Definition:Mapping",
"Inverse Mapping is Bijection",
"Definition:Bijection",
"Definition:Bijection"
] |
proofwiki-10123 | Mapping is Extension iff Composite with Inclusion | Let $S$ and $T$ be sets.
Let $A \subseteq S$.
Let $f: S \to T$ and $g: A \to T$ be mappings.
Then $f$ is an extension of $g$ {{iff}}:
:$f = g \circ i_A$
where $i_A$ is the inclusion mapping on $A$.
This can be illustrated using a commutative diagram as follows:
::<nowiki>$\begin {xy} \xymatrix@L + 2mu@ + 1em {
A \ar[r... | === Necessary Condition ===
Let $f: S \to T$ be an extension of $g: A \to T$.
Then by definition:
{{begin-eqn}}
{{eqn | q = \forall x \in A
| l = \map f x
| r = \map g x
| c =
}}
{{eqn | r = \map g {\map {i_A} x}
| c = {{Defof|Inclusion Mapping}}
}}
{{eqn | r = \map {\paren {g \circ i_A} } x
... | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $A \subseteq S$.
Let $f: S \to T$ and $g: A \to T$ be [[Definition:Mapping|mappings]].
Then $f$ is an [[Definition:Extension of Mapping|extension]] of $g$ {{iff}}:
:$f = g \circ i_A$
where $i_A$ is the [[Definition:Inclusion Mapping|inclusion mapping]] on $A$.
This ... | === Necessary Condition ===
Let $f: S \to T$ be an [[Definition:Extension of Mapping|extension]] of $g: A \to T$.
Then by definition:
{{begin-eqn}}
{{eqn | q = \forall x \in A
| l = \map f x
| r = \map g x
| c =
}}
{{eqn | r = \map g {\map {i_A} x}
| c = {{Defof|Inclusion Mapping}}
}}
{{eqn |... | Mapping is Extension iff Composite with Inclusion | https://proofwiki.org/wiki/Mapping_is_Extension_iff_Composite_with_Inclusion | https://proofwiki.org/wiki/Mapping_is_Extension_iff_Composite_with_Inclusion | [
"Inclusion Mappings",
"Composite Mappings"
] | [
"Definition:Set",
"Definition:Mapping",
"Definition:Extension of Mapping",
"Definition:Inclusion Mapping",
"Definition:Commutative Diagram"
] | [
"Definition:Extension of Mapping",
"Definition:Extension of Mapping"
] |
proofwiki-10124 | Cardinality of Set of Restrictions of Mapping | Let $S$ and $T$ be sets.
Let $f: S \to T$ be a mapping.
Let the cardinality of $S$ be $n$.
Let $F$ be the set of restrictions of $f$ to a subset of $S$.
Then there are $2^n$ elements of $F$. | Let $A \subseteq S$ be a subset of $S$.
Let $g: A \to T$ be the restriction of $f$ to $A$.
By definition of restriction of mapping:
:$\forall x \in A: \map g x = \map f x$
and hence there is one mapping $g: A \to T$ such that $g$ is a restriction of $f$.
Therefore for each subset of $S$ there exists a unique restrictio... | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Let the [[Definition:Cardinality|cardinality]] of $S$ be $n$.
Let $F$ be the [[Definition:Set|set]] of [[Definition:Restriction of Mapping|restrictions]] of $f$ to a [[Definition:Subset|subset]] of $S$.
Then there are... | Let $A \subseteq S$ be a [[Definition:Subset|subset]] of $S$.
Let $g: A \to T$ be the [[Definition:Restriction of Mapping|restriction]] of $f$ to $A$.
By definition of [[Definition:Restriction of Mapping|restriction of mapping]]:
:$\forall x \in A: \map g x = \map f x$
and hence there is one [[Definition:Mapping|map... | Cardinality of Set of Restrictions of Mapping | https://proofwiki.org/wiki/Cardinality_of_Set_of_Restrictions_of_Mapping | https://proofwiki.org/wiki/Cardinality_of_Set_of_Restrictions_of_Mapping | [
"Restrictions"
] | [
"Definition:Set",
"Definition:Mapping",
"Definition:Cardinality",
"Definition:Set",
"Definition:Restriction/Mapping",
"Definition:Subset",
"Definition:Element"
] | [
"Definition:Subset",
"Definition:Restriction/Mapping",
"Definition:Restriction/Mapping",
"Definition:Mapping",
"Definition:Restriction/Mapping",
"Definition:Subset",
"Definition:Unique",
"Definition:Restriction/Mapping",
"Definition:Subset",
"Definition:Restriction/Mapping",
"Definition:Subset",... |
proofwiki-10125 | Projection is Surjection/Family of Sets | Let $\family {S_\alpha}_{\alpha \mathop \in I}$ be a family of sets.
Let $\ds \prod_{\alpha \mathop \in I} S_\alpha$ be the Cartesian product of $\family {S_\alpha}_{\alpha \mathop \in I}$.
Let each of $S_\alpha$ be non-empty.
For each $\beta \in I$, let $\ds \pr_\beta: \prod_{\alpha \mathop \in I} S_\alpha \to S_\beta... | Consider the $\beta$th projection.
Let $x_\beta \in S_\beta$.
Let $\map x \beta = x_\beta$
Suppose $\gamma \in I: \gamma \ne \beta$.
As $S_\gamma \ne \O$ it is possible to use the axiom of choice to choose $\map x \gamma \in S_\gamma$.
Then:
:$\ds x \in \prod_{\alpha \mathop \in I} S_\alpha$
and:
:$\map {\pr_\beta} x =... | Let $\family {S_\alpha}_{\alpha \mathop \in I}$ be a [[Definition:Indexed Family of Sets|family of sets]].
Let $\ds \prod_{\alpha \mathop \in I} S_\alpha$ be the [[Definition:Cartesian Product of Family|Cartesian product]] of $\family {S_\alpha}_{\alpha \mathop \in I}$.
Let each of $S_\alpha$ be [[Definition:Non-Empt... | Consider the $\beta$th projection.
Let $x_\beta \in S_\beta$.
Let $\map x \beta = x_\beta$
Suppose $\gamma \in I: \gamma \ne \beta$.
As $S_\gamma \ne \O$ it is possible to use the [[Axiom:Axiom of Choice|axiom of choice]] to choose $\map x \gamma \in S_\gamma$.
Then:
:$\ds x \in \prod_{\alpha \mathop \in I} S_\alp... | Projection is Surjection/Family of Sets | https://proofwiki.org/wiki/Projection_is_Surjection/Family_of_Sets | https://proofwiki.org/wiki/Projection_is_Surjection/Family_of_Sets | [
"Surjections",
"Indexed Families",
"Projections"
] | [
"Definition:Indexing Set/Family of Sets",
"Definition:Cartesian Product/Family of Sets",
"Definition:Non-Empty Set",
"Definition:Projection (Mapping Theory)/Family of Sets",
"Definition:Surjection"
] | [
"Axiom:Axiom of Choice"
] |
proofwiki-10126 | Taxicab Metric on Real Vector Space is Metric | The taxicab metric on the real vector space $\R^n$ is a metric. | This is an instance of the taxicab metric on the cartesian product of $A_{1'}, A_{2'}, \ldots, A_{n'}$.
This is proved in Taxicab Metric is Metric.
{{qed}} | The [[Definition:Taxicab Metric on Real Vector Space|taxicab metric]] on the [[Definition:Real Vector Space|real vector space]] $\R^n$ is a [[Definition:Metric|metric]]. | This is an instance of the [[Definition:Taxicab Metric/General Definition|taxicab metric]] on the [[Definition:Finite Cartesian Product|cartesian product]] of $A_{1'}, A_{2'}, \ldots, A_{n'}$.
This is proved in [[Taxicab Metric is Metric]].
{{qed}} | Taxicab Metric on Real Vector Space is Metric/Proof 1 | https://proofwiki.org/wiki/Taxicab_Metric_on_Real_Vector_Space_is_Metric | https://proofwiki.org/wiki/Taxicab_Metric_on_Real_Vector_Space_is_Metric/Proof_1 | [
"Taxicab Metric",
"Taxicab Metric on Real Vector Space is Metric"
] | [
"Definition:Taxicab Metric/Real Vector Space",
"Definition:Real Vector Space",
"Definition:Metric Space/Metric"
] | [
"Definition:Taxicab Metric/General Definition",
"Definition:Cartesian Product/Finite",
"Taxicab Metric is Metric"
] |
proofwiki-10127 | Taxicab Metric on Real Vector Space is Metric | The taxicab metric on the real vector space $\R^n$ is a metric. | The taxicab metric on $\R^n$ is:
:$\ds \map {d_1} {x, y} = \sum_{i \mathop = 1}^n \size {x_i - y_i}$
for $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n} \in \R^n$.
=== Proof of {{Metric-space-axiom|1|nolink}} ===
{{begin-eqn}}
{{eqn | l = \map {d_1} {x, x}
| r = \sum_{i \mathop = 1}^n \siz... | The [[Definition:Taxicab Metric on Real Vector Space|taxicab metric]] on the [[Definition:Real Vector Space|real vector space]] $\R^n$ is a [[Definition:Metric|metric]]. | The [[Definition:Taxicab Metric on Real Vector Space|taxicab metric]] on $\R^n$ is:
:$\ds \map {d_1} {x, y} = \sum_{i \mathop = 1}^n \size {x_i - y_i}$
for $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n} \in \R^n$.
=== Proof of {{Metric-space-axiom|1|nolink}} ===
{{begin-eqn}}
{{eqn | l = \ma... | Taxicab Metric on Real Vector Space is Metric/Proof 2 | https://proofwiki.org/wiki/Taxicab_Metric_on_Real_Vector_Space_is_Metric | https://proofwiki.org/wiki/Taxicab_Metric_on_Real_Vector_Space_is_Metric/Proof_2 | [
"Taxicab Metric",
"Taxicab Metric on Real Vector Space is Metric"
] | [
"Definition:Taxicab Metric/Real Vector Space",
"Definition:Real Vector Space",
"Definition:Metric Space/Metric"
] | [
"Definition:Taxicab Metric/Real Vector Space",
"Triangle Inequality/Real Numbers"
] |
proofwiki-10128 | Chebyshev Distance on Real Vector Space is Metric | The Chebyshev distance on $\R^n$:
:$\ds \forall x, y \in \R^n: \map {d_\infty} {x, y}:= \max_{i \mathop = 1}^n {\size {x_i - y_i} }$
is a metric. | This is an instance of the Chebyshev distance on the cartesian product of metric spaces $A_1, A_2, \ldots, A_3$.
This is proved in Chebyshev Distance is Metric.
{{qed}} | The [[Definition:Chebyshev Distance on Real Vector Space|Chebyshev distance]] on $\R^n$:
:$\ds \forall x, y \in \R^n: \map {d_\infty} {x, y}:= \max_{i \mathop = 1}^n {\size {x_i - y_i} }$
is a [[Definition:Metric|metric]]. | This is an instance of the [[Definition:Chebyshev Distance|Chebyshev distance]] on the [[Definition:Finite Cartesian Product|cartesian product]] of [[Definition:Metric Space|metric spaces]] $A_1, A_2, \ldots, A_3$.
This is proved in [[Chebyshev Distance is Metric]].
{{qed}} | Chebyshev Distance on Real Vector Space is Metric/Proof 1 | https://proofwiki.org/wiki/Chebyshev_Distance_on_Real_Vector_Space_is_Metric | https://proofwiki.org/wiki/Chebyshev_Distance_on_Real_Vector_Space_is_Metric/Proof_1 | [
"Chebyshev Distance"
] | [
"Definition:Chebyshev Distance/Real Vector Space",
"Definition:Metric Space/Metric"
] | [
"Definition:Chebyshev Distance",
"Definition:Cartesian Product/Finite",
"Definition:Metric Space",
"Chebyshev Distance is Metric"
] |
proofwiki-10129 | Chebyshev Distance on Real Vector Space is Metric | The Chebyshev distance on $\R^n$:
:$\ds \forall x, y \in \R^n: \map {d_\infty} {x, y}:= \max_{i \mathop = 1}^n {\size {x_i - y_i} }$
is a metric. | === Proof of {{Metric-space-axiom|1|nolink}} ===
{{begin-eqn}}
{{eqn | l = \map {d_\infty} {x, x}
| r = \max_{i \mathop = 1}^n \size {x_i - x_i}
| c = Definition of $d_\infty$
}}
{{eqn | r = 0
| c =
}}
{{end-eqn}}
So {{Metric-space-axiom|1}} holds for $d_\infty$.
{{qed|lemma}}
=== Proof of {{Metric-s... | The [[Definition:Chebyshev Distance on Real Vector Space|Chebyshev distance]] on $\R^n$:
:$\ds \forall x, y \in \R^n: \map {d_\infty} {x, y}:= \max_{i \mathop = 1}^n {\size {x_i - y_i} }$
is a [[Definition:Metric|metric]]. | === Proof of {{Metric-space-axiom|1|nolink}} ===
{{begin-eqn}}
{{eqn | l = \map {d_\infty} {x, x}
| r = \max_{i \mathop = 1}^n \size {x_i - x_i}
| c = Definition of $d_\infty$
}}
{{eqn | r = 0
| c =
}}
{{end-eqn}}
So {{Metric-space-axiom|1}} holds for $d_\infty$.
{{qed|lemma}}
=== Proof of {{Metr... | Chebyshev Distance on Real Vector Space is Metric/Proof 2 | https://proofwiki.org/wiki/Chebyshev_Distance_on_Real_Vector_Space_is_Metric | https://proofwiki.org/wiki/Chebyshev_Distance_on_Real_Vector_Space_is_Metric/Proof_2 | [
"Chebyshev Distance"
] | [
"Definition:Chebyshev Distance/Real Vector Space",
"Definition:Metric Space/Metric"
] | [
"Triangle Inequality/Real Numbers"
] |
proofwiki-10130 | Chebyshev Distance is Metric | 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 \... | === Proof of {{Metric-space-axiom|1|nolink}} ===
{{begin-eqn}}
{{eqn | l = \map {d_\infty} {x, x}
| r = \max_{i \mathop = 1}^n \set {\map {d_i} {x_i, x_i} }
| c = Definition of $d_\infty$
}}
{{eqn | r = 0
| c = as $d_i$ fulfills {{Metric-space-axiom|1}}
}}
{{end-eqn}}
So {{Metric-space-axiom|1}} holds... | 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... | === Proof of {{Metric-space-axiom|1|nolink}} ===
{{begin-eqn}}
{{eqn | l = \map {d_\infty} {x, x}
| r = \max_{i \mathop = 1}^n \set {\map {d_i} {x_i, x_i} }
| c = Definition of $d_\infty$
}}
{{eqn | r = 0
| c = as $d_i$ fulfills {{Metric-space-axiom|1}}
}}
{{end-eqn}}
So {{Metric-space-axiom|1}} hol... | Chebyshev Distance is Metric | https://proofwiki.org/wiki/Chebyshev_Distance_is_Metric | https://proofwiki.org/wiki/Chebyshev_Distance_is_Metric | [
"Chebyshev Distance"
] | [
"Definition:Metric Space",
"Definition:Cartesian Product/Finite",
"Definition:Chebyshev Distance",
"Definition:Metric Space/Metric"
] | [] |
proofwiki-10131 | Positive Multiple of Metric is Metric | Let $M = \struct {A, d}$ be a metric space.
Let $k \in \R_{>0}$ be a (strictly) positive real number.
Let $d_k: A \times A \to \R$ be the function defined as:
:$\forall \tuple {x, y} \in A: \map {d_k} {x, y} = k \cdot \map d {x, y}$
Then $M_k = \struct {A, d_k}$ is a metric space. | === {{Metric-space-axiom|1|nolink}} ===
{{begin-eqn}}
{{eqn | l = \map {d_k} {x, x}
| r = k \cdot \map d {x, x}
| c = Definition of $d_k$
}}
{{eqn | r = 0
| c = as $d$ fulfils {{Metric-space-axiom|1}}
}}
{{end-eqn}}
So {{Metric-space-axiom|1}} holds for $d_k$.
{{qed|lemma}} | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $k \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|(strictly) positive real number]].
Let $d_k: A \times A \to \R$ be the [[Definition:Real-Valued Function|function]] defined as:
:$\forall \tuple {x, y} \in A: \map {d_k} {x, y} =... | === {{Metric-space-axiom|1|nolink}} ===
{{begin-eqn}}
{{eqn | l = \map {d_k} {x, x}
| r = k \cdot \map d {x, x}
| c = Definition of $d_k$
}}
{{eqn | r = 0
| c = as $d$ fulfils {{Metric-space-axiom|1}}
}}
{{end-eqn}}
So {{Metric-space-axiom|1}} holds for $d_k$.
{{qed|lemma}} | Positive Multiple of Metric is Metric | https://proofwiki.org/wiki/Positive_Multiple_of_Metric_is_Metric | https://proofwiki.org/wiki/Positive_Multiple_of_Metric_is_Metric | [
"Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Strictly Positive/Real Number",
"Definition:Real-Valued Function",
"Definition:Metric Space"
] | [] |
proofwiki-10132 | Set of Successive Numbers contains Unique Multiple | Let $m \in \Z_{\ge 1}$.
Then $\set {m, m + 1, \ldots, m + n - 1}$ contains a unique integer that is a multiple of $n$.
That is, in any set containing $n$ successive integers, $n$ divides exactly one of those integers. | Let $S_m = \set {m, m + 1, \ldots, m + n - 1}$ be a set containing $n$ successive integers.
The proof proceeds by induction on $m$, the smallest number in $S_m$. | Let $m \in \Z_{\ge 1}$.
Then $\set {m, m + 1, \ldots, m + n - 1}$ contains a [[Definition:Unique|unique]] [[Definition:Integer|integer]] that is a [[Definition:Multiple of Integer|multiple]] of $n$.
That is, in any [[Definition:Set|set]] containing $n$ successive [[Definition:Integer|integers]], $n$ [[Definition:Divi... | Let $S_m = \set {m, m + 1, \ldots, m + n - 1}$ be a [[Definition:Set|set]] containing $n$ successive [[Definition:Integer|integers]].
The proof proceeds by [[Principle of Mathematical Induction|induction]] on $m$, the smallest number in $S_m$. | Set of Successive Numbers contains Unique Multiple | https://proofwiki.org/wiki/Set_of_Successive_Numbers_contains_Unique_Multiple | https://proofwiki.org/wiki/Set_of_Successive_Numbers_contains_Unique_Multiple | [
"Number Theory"
] | [
"Definition:Unique",
"Definition:Integer",
"Definition:Multiple/Integer",
"Definition:Set",
"Definition:Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Unique",
"Definition:Integer"
] | [
"Definition:Set",
"Definition:Integer",
"Principle of Mathematical Induction",
"Definition:Integer",
"Principle of Mathematical Induction"
] |
proofwiki-10133 | Taxicab Metric is Topologically Equivalent to Chebyshev Distance on Real Vector Space | For $n \in \N$, let $\R^n$ be a real vector space.
Let $d_1$ be the taxicab metric on $\R^n$.
Let $d_\infty$ be the Chebyshev distance on $\R^n$.
Then
:$\forall x, y \in \R^n: \map {d_\infty} {x, y} \le \map {d_1} {x, y} \le n \cdot \map {d_\infty} {x, y}$
It follows that $d_1$ and $d_\infty$ are Lipschitz equivalent. | By definition of the Chebyshev distance on $\R^n$, we have:
:$\ds \map {d_\infty} {x, y} = \max_{i \mathop = 1}^n {\size {x_i - y_i} }$
where $x = \tuple {x_1, x_2, \ldots, x_n}$ and $y = \tuple {y_1, y_2, \ldots, y_n}$.
Let $j$ be chosen so that:
:$\ds \size {x_j - y_j} = \max_{i \mathop = 1}^n {\size {x_i - y_i} }$
T... | For $n \in \N$, let $\R^n$ be a [[Definition:Real Vector Space|real vector space]].
Let $d_1$ be the [[Definition:Taxicab Metric on Real Vector Space|taxicab metric]] on $\R^n$.
Let $d_\infty$ be the [[Definition:Chebyshev Distance on Real Vector Space|Chebyshev distance]] on $\R^n$.
Then
:$\forall x, y \in \R^n: \... | By definition of the [[Definition:Chebyshev Distance on Real Vector Space|Chebyshev distance on $\R^n$]], we have:
:$\ds \map {d_\infty} {x, y} = \max_{i \mathop = 1}^n {\size {x_i - y_i} }$
where $x = \tuple {x_1, x_2, \ldots, x_n}$ and $y = \tuple {y_1, y_2, \ldots, y_n}$.
Let $j$ be chosen so that:
:$\ds \size {x_j... | Taxicab Metric is Topologically Equivalent to Chebyshev Distance on Real Vector Space | https://proofwiki.org/wiki/Taxicab_Metric_is_Topologically_Equivalent_to_Chebyshev_Distance_on_Real_Vector_Space | https://proofwiki.org/wiki/Taxicab_Metric_is_Topologically_Equivalent_to_Chebyshev_Distance_on_Real_Vector_Space | [
"Chebyshev Distance",
"Taxicab Metric"
] | [
"Definition:Real Vector Space",
"Definition:Taxicab Metric/Real Vector Space",
"Definition:Chebyshev Distance/Real Vector Space",
"Definition:Lipschitz Equivalence/Metrics"
] | [
"Definition:Chebyshev Distance/Real Vector Space"
] |
proofwiki-10134 | L1 Metric on Closed Real Interval is Metric | Let $S$ be the set of all real functions which are continuous on the closed interval $\closedint a b$.
Let $d: S \times S \to \R$ be the $L^1$ metric on $\closedint a b$:
:$\ds \forall f, g \in S: \map d {f, g} := \int_a^b \size {\map f t - \map g t} \rd t$
Then $d$ is a metric. | === {{Metric-space-axiom|1|nolink}} ===
{{begin-eqn}}
{{eqn | l = \map d {f, f}
| r = \int_a^b \size {\map f t - \map f t} \rd t
| c = Definition of $d$
}}
{{eqn | r = \int_a^b 0 \rd t
| c = {{Defof|Absolute Value}}
}}
{{eqn | r = 0
| c = Definite Integral of Constant
}}
{{end-eqn}}
So {{Metric-... | Let $S$ be the [[Definition:Set|set]] of all [[Definition:Real Function|real functions]] which are [[Definition:Continuous Real Function|continuous]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$.
Let $d: S \times S \to \R$ be the [[Definition:L1 Metric on Closed Real Interval|$L^1$ metri... | === {{Metric-space-axiom|1|nolink}} ===
{{begin-eqn}}
{{eqn | l = \map d {f, f}
| r = \int_a^b \size {\map f t - \map f t} \rd t
| c = Definition of $d$
}}
{{eqn | r = \int_a^b 0 \rd t
| c = {{Defof|Absolute Value}}
}}
{{eqn | r = 0
| c = [[Definite Integral of Constant]]
}}
{{end-eqn}}
So {{M... | L1 Metric on Closed Real Interval is Metric | https://proofwiki.org/wiki/L1_Metric_on_Closed_Real_Interval_is_Metric | https://proofwiki.org/wiki/L1_Metric_on_Closed_Real_Interval_is_Metric | [
"Definite Integrals",
"L1 Metric"
] | [
"Definition:Set",
"Definition:Real Function",
"Definition:Continuous Real Function",
"Definition:Real Interval/Closed",
"Definition:L1 Metric/Closed Real Interval",
"Definition:Metric Space/Metric"
] | [
"Integral of Constant/Definite"
] |
proofwiki-10135 | Euclidean Metric is Metric | 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 $A_{1'}, A_{2'}, \ldots, A_{n'}$.
The Euclidean metric on $\AA$ is a metric. | The Euclidean metric on $\AA$ is a special case of the $p$-product metric.
The result follows from $p$-Product Metric is Metric.
{{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 $A_{1'}, A_{2'}, \ldots, A_{n'}$.
Th... | The [[Definition:Euclidean Metric|Euclidean metric]] on $\AA$ is a special case of the [[Definition:P-Product Metric|$p$-product metric]].
The result follows from [[P-Product Metric is Metric|$p$-Product Metric is Metric]].
{{qed}} | Euclidean Metric is Metric/Proof 1 | https://proofwiki.org/wiki/Euclidean_Metric_is_Metric | https://proofwiki.org/wiki/Euclidean_Metric_is_Metric/Proof_1 | [
"Euclidean Metric",
"Euclidean Metric is Metric"
] | [
"Definition:Metric Space",
"Definition:Cartesian Product/Finite",
"Definition:Euclidean Metric",
"Definition:Metric Space/Metric"
] | [
"Definition:Euclidean Metric",
"Definition:P-Product Metric",
"P-Product Metric is Metric"
] |
proofwiki-10136 | Euclidean Metric is Metric | 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 $A_{1'}, A_{2'}, \ldots, A_{n'}$.
The Euclidean metric on $\AA$ is a metric. | We have that the Euclidean metric on $\AA$ is defined as:
:$\ds \map {d_2} {x, y} = \paren {\sum_{i \mathop = 1}^n \paren {\map {d_{i'} } {x_i, y_i} }^2}^{\frac 1 2}$
where $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n} \in \AA$.
=== Proof of {{Metric-space-axiom|1|nolink}} ===
{{begin-eqn}}
{{... | 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 $A_{1'}, A_{2'}, \ldots, A_{n'}$.
Th... | We have that the [[Definition:Euclidean Metric|Euclidean metric]] on $\AA$ is defined as:
:$\ds \map {d_2} {x, y} = \paren {\sum_{i \mathop = 1}^n \paren {\map {d_{i'} } {x_i, y_i} }^2}^{\frac 1 2}$
where $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n} \in \AA$.
=== Proof of {{Metric-space-a... | Euclidean Metric is Metric/Proof 2 | https://proofwiki.org/wiki/Euclidean_Metric_is_Metric | https://proofwiki.org/wiki/Euclidean_Metric_is_Metric/Proof_2 | [
"Euclidean Metric",
"Euclidean Metric is Metric"
] | [
"Definition:Metric Space",
"Definition:Cartesian Product/Finite",
"Definition:Euclidean Metric",
"Definition:Metric Space/Metric"
] | [
"Definition:Euclidean Metric",
"Minkowski's Inequality for Sums/Index 2"
] |
proofwiki-10137 | Euclidean Space is Normed Vector Space | The real Euclidean space $\R^n$ is a normed vector space. | Let $\norm {\, \cdot \,}$ denote the Euclidean norm on $\R^n$.
We prove that $\norm {\, \cdot \,}$ is indeed a norm on $\R^n$ by proving it fulfils the norm axioms. | The [[Definition:Real Euclidean Space|real Euclidean space]] $\R^n$ is a [[Definition:Normed Vector Space|normed vector space]]. | Let $\norm {\, \cdot \,}$ denote the [[Definition:Euclidean Norm|Euclidean norm]] on $\R^n$.
We prove that $\norm {\, \cdot \,}$ is indeed a [[Definition:Norm on Vector Space|norm]] on $\R^n$ by proving it fulfils the [[Axiom:Vector Space Norm Axioms|norm axioms]]. | Euclidean Space is Normed Vector Space | https://proofwiki.org/wiki/Euclidean_Space_is_Normed_Vector_Space | https://proofwiki.org/wiki/Euclidean_Space_is_Normed_Vector_Space | [
"Real Euclidean Spaces",
"Examples of Norms"
] | [
"Definition:Euclidean Space/Real",
"Definition:Normed Vector Space"
] | [
"Definition:Euclidean Norm",
"Definition:Norm/Vector Space",
"Axiom:Vector Space Norm Axioms"
] |
proofwiki-10138 | Number of Multiples less than Given Number | Let $m, n \in \N_{\ge 1}$.
The number of multiples of $m$ not greater than $n$ is given by:
:$q = \floor {\dfrac n m}$
where $\floor {\cdot}$ denotes the floor function | By the Division Theorem:
:$(1): \quad n = q m + r$
where $0 \le r < q$.
As $r < q$, it follows that the greatest multiple of $m$ up to $n$ is $q m$.
So all the multiples of $m$ up to $n$ are:
:$m, 2 m, 3 m, \ldots, q m$
Dividing both sides of $(1)$ by $m$:
:$(2): \quad \dfrac n m = q + \dfrac r m$
Taking the floor of $... | Let $m, n \in \N_{\ge 1}$.
The number of [[Definition:Multiple of Integer|multiples]] of $m$ not greater than $n$ is given by:
:$q = \floor {\dfrac n m}$
where $\floor {\cdot}$ denotes the [[Definition:Floor Function|floor function]] | By the [[Division Theorem]]:
:$(1): \quad n = q m + r$
where $0 \le r < q$.
As $r < q$, it follows that the greatest [[Definition:Multiple of Integer|multiple]] of $m$ up to $n$ is $q m$.
So all the [[Definition:Multiple of Integer|multiples]] of $m$ up to $n$ are:
:$m, 2 m, 3 m, \ldots, q m$
Dividing both sides o... | Number of Multiples less than Given Number | https://proofwiki.org/wiki/Number_of_Multiples_less_than_Given_Number | https://proofwiki.org/wiki/Number_of_Multiples_less_than_Given_Number | [
"Number Theory"
] | [
"Definition:Multiple/Integer",
"Definition:Floor Function"
] | [
"Division Theorem",
"Definition:Multiple/Integer",
"Definition:Multiple/Integer",
"Definition:Floor Function",
"Definition:Multiple/Integer",
"Definition:Multiple/Integer"
] |
proofwiki-10139 | Restriction of Non-Continuous Mapping on Metric Space to Subspace may be Continuous | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be a metric spaces.
Let $f: A_1 \to A_2$ be a mapping.
Let $Y \subseteq A_1$.
Let $f {\restriction_Y}: Y \to A_2$ be the restriction of $f$ to $Y$.
Let $f {\restriction_Y}$ be $\tuple {d_Y, d_2}$-continuous.
Then it is not necessarily the case that $f$ is al... | Proof by Counterexample:
Let $f: \R \to \R$ be given by:
:<nowiki>$\map f x = \begin {cases}
0 & : x \in \Q \\
1 & : x \in \R \setminus \Q
\end {cases}$</nowiki>
where $\Q$ is the set of rational numbers.
Then $f {\restriction_\Q}: \Q \to \R$ is the constant function $f_0$ with value $0$.
By Constant Mapping is Continu... | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be a [[Definition:Metric Space|metric spaces]].
Let $f: A_1 \to A_2$ be a [[Definition:Mapping|mapping]].
Let $Y \subseteq A_1$.
Let $f {\restriction_Y}: Y \to A_2$ be the [[Definition:Restriction of Mapping|restriction]] of $f$ to $Y$.
Let $f {\restric... | [[Proof by Counterexample]]:
Let $f: \R \to \R$ be given by:
:<nowiki>$\map f x = \begin {cases}
0 & : x \in \Q \\
1 & : x \in \R \setminus \Q
\end {cases}$</nowiki>
where $\Q$ is the set of [[Definition:Rational Number|rational numbers]].
Then $f {\restriction_\Q}: \Q \to \R$ is the [[Definition:Constant Mapping|c... | Restriction of Non-Continuous Mapping on Metric Space to Subspace may be Continuous | https://proofwiki.org/wiki/Restriction_of_Non-Continuous_Mapping_on_Metric_Space_to_Subspace_may_be_Continuous | https://proofwiki.org/wiki/Restriction_of_Non-Continuous_Mapping_on_Metric_Space_to_Subspace_may_be_Continuous | [
"Continuity",
"Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Mapping",
"Definition:Restriction/Mapping",
"Definition:Continuous Mapping/Metric Subspace",
"Definition:Continuous Mapping/Metric Subspace"
] | [
"Proof by Counterexample",
"Definition:Rational Number",
"Definition:Constant Mapping",
"Constant Mapping is Continuous",
"Definition:Continuous Mapping/Metric Subspace",
"Definition:Continuous Real Function/Point"
] |
proofwiki-10140 | Supremum Metric on Bounded Real Functions on Closed Interval is Metric | Let $\closedint a b \subseteq \R$ be a closed real interval.
Let $A$ be the set of all bounded real functions $f: \closedint a b \to \R$.
Let $d: A \times A \to \R$ be the supremum metric on $A$.
Then $d$ is a metric. | The interval is an instance of a set.
Hence Supremum Metric on Bounded Real-Valued Functions is Metric can be directly applied. | Let $\closedint a b \subseteq \R$ be a [[Definition:Closed Real Interval|closed real interval]].
Let $A$ be the [[Definition:Set|set]] of all [[Definition:Bounded Real-Valued Function|bounded]] [[Definition:Real Function|real functions]] $f: \closedint a b \to \R$.
Let $d: A \times A \to \R$ be the [[Definition:Supre... | The [[Definition:Closed Real Interval|interval]] is an instance of a [[Definition:Set|set]].
Hence [[Supremum Metric on Bounded Real-Valued Functions is Metric]] can be directly applied. | Supremum Metric on Bounded Real Functions on Closed Interval is Metric/Proof 1 | https://proofwiki.org/wiki/Supremum_Metric_on_Bounded_Real_Functions_on_Closed_Interval_is_Metric | https://proofwiki.org/wiki/Supremum_Metric_on_Bounded_Real_Functions_on_Closed_Interval_is_Metric/Proof_1 | [
"Supremum Metric",
"Supremum Metric on Bounded Real Functions on Closed Interval is Metric"
] | [
"Definition:Real Interval/Closed",
"Definition:Set",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Real Function",
"Definition:Supremum Metric/Bounded Real Functions on Interval",
"Definition:Metric Space/Metric"
] | [
"Definition:Real Interval/Closed",
"Definition:Set",
"Supremum Metric on Bounded Real-Valued Functions is Metric"
] |
proofwiki-10141 | Supremum Metric on Bounded Real Functions on Closed Interval is Metric | Let $\closedint a b \subseteq \R$ be a closed real interval.
Let $A$ be the set of all bounded real functions $f: \closedint a b \to \R$.
Let $d: A \times A \to \R$ be the supremum metric on $A$.
Then $d$ is a metric. | We have that the supremum metric on $A \times A$ is defined as:
:$\ds \forall f, g \in A: \map d {f, g} := \sup_{x \mathop \in \closedint a b} \size {\map f x - \map g x}$
where $f$ and $g$ are bounded real functions.
So:
:$\exists K, L \in \R: \size {\map f x} \le K, \size {\map g x} \le L$
for all $x \in \closedint a... | Let $\closedint a b \subseteq \R$ be a [[Definition:Closed Real Interval|closed real interval]].
Let $A$ be the [[Definition:Set|set]] of all [[Definition:Bounded Real-Valued Function|bounded]] [[Definition:Real Function|real functions]] $f: \closedint a b \to \R$.
Let $d: A \times A \to \R$ be the [[Definition:Supre... | We have that the [[Definition:Supremum Metric on Bounded Real Functions on Closed Interval|supremum metric]] on $A \times A$ is defined as:
:$\ds \forall f, g \in A: \map d {f, g} := \sup_{x \mathop \in \closedint a b} \size {\map f x - \map g x}$
where $f$ and $g$ are [[Definition:Bounded Real-Valued Function|bounde... | Supremum Metric on Bounded Real Functions on Closed Interval is Metric/Proof 2 | https://proofwiki.org/wiki/Supremum_Metric_on_Bounded_Real_Functions_on_Closed_Interval_is_Metric | https://proofwiki.org/wiki/Supremum_Metric_on_Bounded_Real_Functions_on_Closed_Interval_is_Metric/Proof_2 | [
"Supremum Metric",
"Supremum Metric on Bounded Real Functions on Closed Interval is Metric"
] | [
"Definition:Real Interval/Closed",
"Definition:Set",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Real Function",
"Definition:Supremum Metric/Bounded Real Functions on Interval",
"Definition:Metric Space/Metric"
] | [
"Definition:Supremum Metric/Bounded Real Functions on Interval",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Real Function",
"Triangle Inequality/Real Numbers",
"Triangle Inequality/Real Numbers",
"Definition:Upper Bound of Mapping/Real-Valued",
"Definition:Supremum of Mapping/Real-Valued Func... |
proofwiki-10142 | Supremum Metric on Continuous Real Functions is Subspace of Bounded | Let $\closedint a b \subseteq \R$ be a closed real interval.
Let $\mathscr C \closedint a b$ be the set of all continuous functions $f: \closedint a b \to \R$.
Let $\map {\mathscr B} {\closedint a b, \R}$ be the set of all bounded real functions $f: \closedint a b \to \R$.
Let $d$ be the supremum metric on $\map {\math... | Let $f \in \mathscr C \closedint a b$.
Then by Image of Closed Real Interval is Bounded, $f$ is bounded on $\closedint a b$.
Thus $f \in \map {\mathscr B} {\closedint a b, \R}$ and the result follows.
{{qed}} | Let $\closedint a b \subseteq \R$ be a [[Definition:Closed Real Interval|closed real interval]].
Let $\mathscr C \closedint a b$ be the [[Definition:Set|set]] of all [[Definition:Continuous Real Function on Closed Interval|continuous functions]] $f: \closedint a b \to \R$.
Let $\map {\mathscr B} {\closedint a b, \R}$... | Let $f \in \mathscr C \closedint a b$.
Then by [[Image of Closed Real Interval is Bounded]], $f$ is [[Definition:Bounded Real-Valued Function|bounded]] on $\closedint a b$.
Thus $f \in \map {\mathscr B} {\closedint a b, \R}$ and the result follows.
{{qed}} | Supremum Metric on Continuous Real Functions is Subspace of Bounded | https://proofwiki.org/wiki/Supremum_Metric_on_Continuous_Real_Functions_is_Subspace_of_Bounded | https://proofwiki.org/wiki/Supremum_Metric_on_Continuous_Real_Functions_is_Subspace_of_Bounded | [
"Supremum Metric"
] | [
"Definition:Real Interval/Closed",
"Definition:Set",
"Definition:Continuous Real Function/Closed Interval",
"Definition:Set",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Real Function",
"Definition:Supremum Metric/Bounded Real Functions on Interval",
"Definition:Metric Subspace"
] | [
"Image of Closed Real Interval is Bounded",
"Definition:Bounded Mapping/Real-Valued"
] |
proofwiki-10143 | Zero Definite Integral of Nowhere Negative Function implies Zero Function | Let $\closedint a b \subseteq \R$ be a closed real interval.
Let $h: \closedint a b \to \R$ be a continuous real function such that:
:$\forall x \in \closedint a b: \map h x \ge 0$
Let:
:$\ds \int_a^b \map h x \rd x = 0$
Then:
:$\forall x \in \closedint a b: \map h x = 0$ | {{AimForCont}} that:
:$\exists c \in \closedint a b: \map h c > 0$
As $h$ is continuous, there exists some closed real interval $\closedint r s \subseteq \closedint a b$ where $r < s$ such that:
:$\exists \epsilon \in \R_{>0}: \forall x \in \closedint r s: \map h x > \dfrac {\map h c} 2$
From Sign of Function Matches S... | Let $\closedint a b \subseteq \R$ be a [[Definition:Closed Real Interval|closed real interval]].
Let $h: \closedint a b \to \R$ be a [[Definition:Continuous Real Function on Closed Interval|continuous]] [[Definition:Real Function|real function]] such that:
:$\forall x \in \closedint a b: \map h x \ge 0$
Let:
:$\ds \i... | {{AimForCont}} that:
:$\exists c \in \closedint a b: \map h c > 0$
As $h$ is [[Definition:Continuous Real Function on Closed Interval|continuous]], there exists some [[Definition:Closed Real Interval|closed real interval]] $\closedint r s \subseteq \closedint a b$ where $r < s$ such that:
:$\exists \epsilon \in \R_{>0... | Zero Definite Integral of Nowhere Negative Function implies Zero Function | https://proofwiki.org/wiki/Zero_Definite_Integral_of_Nowhere_Negative_Function_implies_Zero_Function | https://proofwiki.org/wiki/Zero_Definite_Integral_of_Nowhere_Negative_Function_implies_Zero_Function | [
"Definite Integrals"
] | [
"Definition:Real Interval/Closed",
"Definition:Continuous Real Function/Closed Interval",
"Definition:Real Function"
] | [
"Definition:Continuous Real Function/Closed Interval",
"Definition:Real Interval/Closed",
"Sign of Function Matches Sign of Definite Integral",
"Definition:Strictly Positive/Real Number",
"Definition:Definite Integral",
"Definition:Continuous Real Function/Closed Interval",
"Sign of Function Matches Sig... |
proofwiki-10144 | L2 Metric on Closed Real Interval is Metric | Let $S$ be the set of all real functions which are continuous on the closed interval $\closedint a b$.
Let $d_2: S \times S \to \R$ be the $L^2$ metric on $\closedint a b$:
:$\ds \forall f, g \in S: \map {d_2} {f, g} := \paren {\int_a^b \paren {\map f t - \map g t}^2 \rd t}^{\frac 1 2}$
Then $d_2$ is a metric. | === Proof of {{Metric-space-axiom|1|nolink}} ===
{{begin-eqn}}
{{eqn | l = \map {d_2} {f, f}
| r = \paren {\int_a^b \paren {\map f t - \map f t}^2 \rd t}^{\frac 1 2}
| c = Definition of $d_2$
}}
{{eqn | r = \paren {\int_a^b 0^2 \rd t}^{\frac 1 2}
| c = {{Defof|Absolute Value}}
}}
{{eqn | r = 0
|... | Let $S$ be the [[Definition:Set|set]] of all [[Definition:Real Function|real functions]] which are [[Definition:Continuous Real Function|continuous]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$.
Let $d_2: S \times S \to \R$ be the [[Definition:L2 Metric on Closed Real Interval|$L^2$ met... | === Proof of {{Metric-space-axiom|1|nolink}} ===
{{begin-eqn}}
{{eqn | l = \map {d_2} {f, f}
| r = \paren {\int_a^b \paren {\map f t - \map f t}^2 \rd t}^{\frac 1 2}
| c = Definition of $d_2$
}}
{{eqn | r = \paren {\int_a^b 0^2 \rd t}^{\frac 1 2}
| c = {{Defof|Absolute Value}}
}}
{{eqn | r = 0
... | L2 Metric on Closed Real Interval is Metric | https://proofwiki.org/wiki/L2_Metric_on_Closed_Real_Interval_is_Metric | https://proofwiki.org/wiki/L2_Metric_on_Closed_Real_Interval_is_Metric | [
"L2 Metric"
] | [
"Definition:Set",
"Definition:Real Function",
"Definition:Continuous Real Function",
"Definition:Real Interval/Closed",
"Definition:L2 Metric/Closed Real Interval",
"Definition:Metric Space/Metric"
] | [
"Integral of Constant/Definite"
] |
proofwiki-10145 | Element in Bounded Metric Space has Bound | Let $M = \struct {X, d}$ be a metric space.
Let $M' = \struct {Y, d_Y}$ be a subspace of $M$.
Let $M'$ be bounded in $M$.
Then:
:$\forall a' \in X: \exists K' \in \R: \forall x \in Y: \map d {x, a'} \le K'$
That is, if there is one element of $X$ which satisfies the condition for $Y$ to be bounded in $M$, they ''all'' ... | Let $a \in X$ such that $\exists K \in \R: \forall x \in Y: \map d {x, a} \le K$.
Let $a' \in X$.
{{begin-eqn}}
{{eqn | l = \map d {x, a'}
| o = \le
| r = \map d {x, a} + \map d {a, a'}
| c = {{Metric-space-axiom|2}}
}}
{{eqn | o = \le
| r = K + \map d {a, a'}
| c = by hypothesis
}}
{{eqn ... | Let $M = \struct {X, d}$ be a [[Definition:Metric Space|metric space]].
Let $M' = \struct {Y, d_Y}$ be a [[Definition:Metric Subspace|subspace]] of $M$.
Let $M'$ be [[Definition:Bounded Metric Space|bounded in $M$]].
Then:
:$\forall a' \in X: \exists K' \in \R: \forall x \in Y: \map d {x, a'} \le K'$
That is, if t... | Let $a \in X$ such that $\exists K \in \R: \forall x \in Y: \map d {x, a} \le K$.
Let $a' \in X$.
{{begin-eqn}}
{{eqn | l = \map d {x, a'}
| o = \le
| r = \map d {x, a} + \map d {a, a'}
| c = {{Metric-space-axiom|2}}
}}
{{eqn | o = \le
| r = K + \map d {a, a'}
| c = [[Definition:By Hypot... | Element in Bounded Metric Space has Bound | https://proofwiki.org/wiki/Element_in_Bounded_Metric_Space_has_Bound | https://proofwiki.org/wiki/Element_in_Bounded_Metric_Space_has_Bound | [
"Bounded Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Metric Subspace",
"Definition:Bounded Metric Space",
"Definition:Element",
"Definition:Bounded Metric Space"
] | [
"Definition:By Hypothesis"
] |
proofwiki-10146 | Convergence of Square of Linear Combination of Sequences whose Squares Converge | Let $\sequence {x_i}$ and $\sequence {y_i}$ be real sequences such that the series $\ds \sum_{i \mathop \ge 0} {x_i}^2$ and $\ds \sum_{i \mathop \ge 0} {y_i}^2$ are convergent.
Let $\lambda, \mu \in \R$ be real numbers.
Then $\ds \sum_{i \mathop \ge 0} \paren {\lambda x_i + \mu y_i}^2$ is convergent. | Let $n \in \N$.
Then:
:$\ds \sum_{i \mathop = 1}^n \paren {\lambda x_i + \mu y_i}^2 = \lambda^2 \sum_{i \mathop = 1}^n {x_i}^2 + \mu^2 \sum_{i \mathop = 1}^n {y_i}^2 + 2 \lambda \mu \sum_{i \mathop = 1}^n x_i y_i$
By Cauchy's Inequality:
:$\ds \sum_{i \mathop = 1}^n x_i y_i \le \paren {\sum_{i \mathop = 1}^n {x_i}^2}^{... | Let $\sequence {x_i}$ and $\sequence {y_i}$ be [[Definition:Real Sequence|real sequences]] such that the [[Definition:Series of Numbers|series]] $\ds \sum_{i \mathop \ge 0} {x_i}^2$ and $\ds \sum_{i \mathop \ge 0} {y_i}^2$ are [[Definition:Convergent Series of Numbers|convergent]].
Let $\lambda, \mu \in \R$ be [[Defin... | Let $n \in \N$.
Then:
:$\ds \sum_{i \mathop = 1}^n \paren {\lambda x_i + \mu y_i}^2 = \lambda^2 \sum_{i \mathop = 1}^n {x_i}^2 + \mu^2 \sum_{i \mathop = 1}^n {y_i}^2 + 2 \lambda \mu \sum_{i \mathop = 1}^n x_i y_i$
By [[Cauchy's Inequality]]:
:$\ds \sum_{i \mathop = 1}^n x_i y_i \le \paren {\sum_{i \mathop = 1}^n {x_i... | Convergence of Square of Linear Combination of Sequences whose Squares Converge | https://proofwiki.org/wiki/Convergence_of_Square_of_Linear_Combination_of_Sequences_whose_Squares_Converge | https://proofwiki.org/wiki/Convergence_of_Square_of_Linear_Combination_of_Sequences_whose_Squares_Converge | [
"Real Analysis",
"Series"
] | [
"Definition:Real Sequence",
"Definition:Series/Number Field",
"Definition:Convergent Series/Number Field",
"Definition:Real Number",
"Definition:Convergent Series/Number Field"
] | [
"Cauchy's Inequality",
"Definition:Series/Sequence of Partial Sums",
"Definition:Bounded Above Sequence/Real",
"Definition:Increasing/Sequence/Real Sequence",
"Monotone Convergence Theorem (Real Analysis)",
"Definition:Convergent Series/Number Field"
] |
proofwiki-10147 | Hilbert Sequence Space is Metric Space | Let $A$ be the set of all real sequences $\sequence {x_i}$ such that the series $\ds \sum_{i \mathop \ge 0} {x_i}^2$ is convergent.
Let $\ell^2 = \struct {A, d_2}$ be the Hilbert sequence space on $\R$.
Then $\ell^2$ is a metric space. | $\ell^2$ is a particular instance of the general $p$-sequence space $\ell^p$.
Hence $p$-Sequence Space of Real Sequences is Metric Space can be applied directly.
{{qed}} | Let $A$ be the [[Definition:Set|set]] of all [[Definition:Real Sequence|real sequences]] $\sequence {x_i}$ such that the [[Definition:Series of Numbers|series]] $\ds \sum_{i \mathop \ge 0} {x_i}^2$ is [[Definition:Convergent Series of Numbers|convergent]].
Let $\ell^2 = \struct {A, d_2}$ be the [[Definition:Hilbert Se... | $\ell^2$ is a particular instance of the general [[Definition:P-Sequence Metric|$p$-sequence space]] $\ell^p$.
Hence [[P-Sequence Space of Real Sequences is Metric Space|$p$-Sequence Space of Real Sequences is Metric Space]] can be applied directly.
{{qed}} | Hilbert Sequence Space is Metric Space/Proof 1 | https://proofwiki.org/wiki/Hilbert_Sequence_Space_is_Metric_Space | https://proofwiki.org/wiki/Hilbert_Sequence_Space_is_Metric_Space/Proof_1 | [
"Hilbert Sequence Space is Metric Space",
"Hilbert Sequence Space"
] | [
"Definition:Set",
"Definition:Real Sequence",
"Definition:Series/Number Field",
"Definition:Convergent Series/Number Field",
"Definition:Hilbert Sequence Space",
"Definition:Metric Space"
] | [
"Definition:P-Sequence Metric",
"P-Sequence Space of Real Sequences is Metric Space"
] |
proofwiki-10148 | Hilbert Sequence Space is Metric Space | Let $A$ be the set of all real sequences $\sequence {x_i}$ such that the series $\ds \sum_{i \mathop \ge 0} {x_i}^2$ is convergent.
Let $\ell^2 = \struct {A, d_2}$ be the Hilbert sequence space on $\R$.
Then $\ell^2$ is a metric space. | By definition of the Hilbert sequence space on $\R$:
Let $A$ be the set of all real sequences $\sequence {x_i}$ such that the series $\ds \sum_{i \mathop \ge 0} {x_i}^2$ is convergent.
Then $\ell^2 := \struct {A, d_2}$ where $d_2: A \times A: \to \R$ is the real-valued function defined as:
:$\ds \forall x = \sequence {... | Let $A$ be the [[Definition:Set|set]] of all [[Definition:Real Sequence|real sequences]] $\sequence {x_i}$ such that the [[Definition:Series of Numbers|series]] $\ds \sum_{i \mathop \ge 0} {x_i}^2$ is [[Definition:Convergent Series of Numbers|convergent]].
Let $\ell^2 = \struct {A, d_2}$ be the [[Definition:Hilbert Se... | By definition of the [[Definition:Hilbert Sequence Space|Hilbert sequence space on $\R$]]:
Let $A$ be the [[Definition:Set|set]] of all [[Definition:Real Sequence|real sequences]] $\sequence {x_i}$ such that the [[Definition:Series of Numbers|series]] $\ds \sum_{i \mathop \ge 0} {x_i}^2$ is [[Definition:Convergent Ser... | Hilbert Sequence Space is Metric Space/Proof 2 | https://proofwiki.org/wiki/Hilbert_Sequence_Space_is_Metric_Space | https://proofwiki.org/wiki/Hilbert_Sequence_Space_is_Metric_Space/Proof_2 | [
"Hilbert Sequence Space is Metric Space",
"Hilbert Sequence Space"
] | [
"Definition:Set",
"Definition:Real Sequence",
"Definition:Series/Number Field",
"Definition:Convergent Series/Number Field",
"Definition:Hilbert Sequence Space",
"Definition:Metric Space"
] | [
"Definition:Hilbert Sequence Space",
"Definition:Set",
"Definition:Real Sequence",
"Definition:Series/Number Field",
"Definition:Convergent Series/Number Field",
"Definition:Real-Valued Function",
"Convergence of Square of Linear Combination of Sequences whose Squares Converge",
"Minkowski's Inequalit... |
proofwiki-10149 | Supremum Metric on Bounded Real-Valued Functions is Metric | Let $X$ be a set.
Let $A$ be the set of all bounded real-valued functions $f: X \to \R$.
Let $d: A \times A \to \R$ be the supremum metric on $A$.
Then $d$ is a metric. | We have that the supremum metric on $A \times A$ is defined as:
:$\ds \forall f, g \in A: \map d {f, g} := \sup_{x \mathop \in X} \size {\map f x - \map g x}$
where $f$ and $g$ are bounded real-valued functions.
From Real Number Line is Metric Space, the real numbers $\R$ together with the absolute value function form ... | Let $X$ be a [[Definition:Set|set]].
Let $A$ be the [[Definition:Set|set]] of all [[Definition:Bounded Real-Valued Function|bounded real-valued functions]] $f: X \to \R$.
Let $d: A \times A \to \R$ be the [[Definition:Supremum Metric on Bounded Real-Valued Functions|supremum metric]] on $A$.
Then $d$ is a [[Definit... | We have that the [[Definition:Supremum Metric on Bounded Real-Valued Functions|supremum metric]] on $A \times A$ is defined as:
:$\ds \forall f, g \in A: \map d {f, g} := \sup_{x \mathop \in X} \size {\map f x - \map g x}$
where $f$ and $g$ are [[Definition:Bounded Real-Valued Function|bounded real-valued functions]]... | Supremum Metric on Bounded Real-Valued Functions is Metric/Proof 1 | https://proofwiki.org/wiki/Supremum_Metric_on_Bounded_Real-Valued_Functions_is_Metric | https://proofwiki.org/wiki/Supremum_Metric_on_Bounded_Real-Valued_Functions_is_Metric/Proof_1 | [
"Supremum Metric",
"Supremum Metric on Bounded Real-Valued Functions is Metric"
] | [
"Definition:Set",
"Definition:Set",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Supremum Metric/Bounded Real-Valued Functions",
"Definition:Metric Space/Metric"
] | [
"Definition:Supremum Metric/Bounded Real-Valued Functions",
"Definition:Bounded Mapping/Real-Valued",
"Real Number Line is Metric Space",
"Definition:Real Number",
"Definition:Absolute Value",
"Definition:Metric Space",
"Supremum Metric is Metric"
] |
proofwiki-10150 | Supremum Metric on Bounded Real-Valued Functions is Metric | Let $X$ be a set.
Let $A$ be the set of all bounded real-valued functions $f: X \to \R$.
Let $d: A \times A \to \R$ be the supremum metric on $A$.
Then $d$ is a metric. | We have that the supremum metric on $A \times A$ is defined as:
:$\ds \forall f, g \in A: \map d {f, g} := \sup_{x \mathop \in X} \size {\map f x - \map g x}$
where $f$ and $g$ are bounded real-valued functions.
So:
:$\exists K, L \in \R: \size {\map f x} \le K, \size {\map g x} \le L$
for all $x \in X$.
First note tha... | Let $X$ be a [[Definition:Set|set]].
Let $A$ be the [[Definition:Set|set]] of all [[Definition:Bounded Real-Valued Function|bounded real-valued functions]] $f: X \to \R$.
Let $d: A \times A \to \R$ be the [[Definition:Supremum Metric on Bounded Real-Valued Functions|supremum metric]] on $A$.
Then $d$ is a [[Definit... | We have that the [[Definition:Supremum Metric on Bounded Real-Valued Functions|supremum metric]] on $A \times A$ is defined as:
:$\ds \forall f, g \in A: \map d {f, g} := \sup_{x \mathop \in X} \size {\map f x - \map g x}$
where $f$ and $g$ are [[Definition:Bounded Real-Valued Function|bounded real-valued functions]]... | Supremum Metric on Bounded Real-Valued Functions is Metric/Proof 2 | https://proofwiki.org/wiki/Supremum_Metric_on_Bounded_Real-Valued_Functions_is_Metric | https://proofwiki.org/wiki/Supremum_Metric_on_Bounded_Real-Valued_Functions_is_Metric/Proof_2 | [
"Supremum Metric",
"Supremum Metric on Bounded Real-Valued Functions is Metric"
] | [
"Definition:Set",
"Definition:Set",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Supremum Metric/Bounded Real-Valued Functions",
"Definition:Metric Space/Metric"
] | [
"Definition:Supremum Metric/Bounded Real-Valued Functions",
"Definition:Bounded Mapping/Real-Valued",
"Triangle Inequality/Real Numbers",
"Triangle Inequality/Real Numbers",
"Definition:Upper Bound of Mapping/Real-Valued",
"Definition:Supremum of Mapping/Real-Valued Function",
"Definition:Absolute Value... |
proofwiki-10151 | Supremum Metric on Bounded Real Sequences is Metric | Let $A$ be the set of all bounded real sequences.
Let $d: A \times A \to \R$ be the supremum metric on $A$.
Then $d$ is a metric. | By definition, a real sequence is a mapping from the natural numbers $\N$ to the real numbers $\R$.
Thus a bounded real sequence is a bounded real-valued function.
The result follows from Supremum Metric on Bounded Real-Valued Functions is Metric.
{{qed}} | Let $A$ be the [[Definition:Set|set]] of all [[Definition:Bounded Real Sequence|bounded real sequences]].
Let $d: A \times A \to \R$ be the [[Definition:Supremum Metric on Bounded Real Sequences|supremum metric]] on $A$.
Then $d$ is a [[Definition:Metric|metric]]. | By definition, a [[Definition:Real Sequence|real sequence]] is a [[Definition:Mapping|mapping]] from the [[Definition:Natural Numbers|natural numbers]] $\N$ to the [[Definition:Real Number|real numbers]] $\R$.
Thus a [[Definition:Bounded Real Sequence|bounded real sequence]] is a [[Definition:Bounded Real-Valued Funct... | Supremum Metric on Bounded Real Sequences is Metric/Proof 1 | https://proofwiki.org/wiki/Supremum_Metric_on_Bounded_Real_Sequences_is_Metric | https://proofwiki.org/wiki/Supremum_Metric_on_Bounded_Real_Sequences_is_Metric/Proof_1 | [
"Supremum Metric",
"Supremum Metric on Bounded Real Sequences is Metric"
] | [
"Definition:Set",
"Definition:Bounded Sequence/Real",
"Definition:Supremum Metric/Bounded Real Sequences",
"Definition:Metric Space/Metric"
] | [
"Definition:Real Sequence",
"Definition:Mapping",
"Definition:Natural Numbers",
"Definition:Real Number",
"Definition:Bounded Sequence/Real",
"Definition:Bounded Mapping/Real-Valued",
"Supremum Metric on Bounded Real-Valued Functions is Metric"
] |
proofwiki-10152 | Supremum Metric on Bounded Real Sequences is Metric | Let $A$ be the set of all bounded real sequences.
Let $d: A \times A \to \R$ be the supremum metric on $A$.
Then $d$ is a metric. | We have that the supremum metric on $A \times A$ is defined as:
:$\ds \forall x, y \in A: \map d {x, y} := \sup_{n \mathop \in \N} \size {x_n - y_n}$
where $x = \sequence {x_i}$ and $y = \sequence {y_i}$ are bounded real sequences.
So:
:$\exists K, L \in \R: \size {x_n} \le K, \size {y_n} \le L$
for all $n \in \N$.
Fir... | Let $A$ be the [[Definition:Set|set]] of all [[Definition:Bounded Real Sequence|bounded real sequences]].
Let $d: A \times A \to \R$ be the [[Definition:Supremum Metric on Bounded Real Sequences|supremum metric]] on $A$.
Then $d$ is a [[Definition:Metric|metric]]. | We have that the [[Definition:Supremum Metric on Bounded Real Sequences|supremum metric]] on $A \times A$ is defined as:
:$\ds \forall x, y \in A: \map d {x, y} := \sup_{n \mathop \in \N} \size {x_n - y_n}$
where $x = \sequence {x_i}$ and $y = \sequence {y_i}$ are [[Definition:Bounded Real Sequence|bounded real seque... | Supremum Metric on Bounded Real Sequences is Metric/Proof 2 | https://proofwiki.org/wiki/Supremum_Metric_on_Bounded_Real_Sequences_is_Metric | https://proofwiki.org/wiki/Supremum_Metric_on_Bounded_Real_Sequences_is_Metric/Proof_2 | [
"Supremum Metric",
"Supremum Metric on Bounded Real Sequences is Metric"
] | [
"Definition:Set",
"Definition:Bounded Sequence/Real",
"Definition:Supremum Metric/Bounded Real Sequences",
"Definition:Metric Space/Metric"
] | [
"Definition:Supremum Metric/Bounded Real Sequences",
"Definition:Bounded Sequence/Real",
"Triangle Inequality/Real Numbers",
"Triangle Inequality/Real Numbers",
"Definition:Upper Bound of Mapping/Real-Valued",
"Definition:Supremum of Real Sequence",
"Definition:Absolute Value",
"Definition:Term of Seq... |
proofwiki-10153 | Supremum Metric is Metric | Let $S$ be a set.
Let $M = \struct {A', d'}$ be a metric space.
Let $A$ be the set of all bounded mappings $f: S \to M$.
Let $d: A \times A \to \R$ be the supremum metric on $A$.
Then $d$ is a metric. | We have that the supremum metric on $A \times A$ is defined as:
:$\ds \forall f, g \in A: \map d {f, g} := \sup_{x \mathop \in S} \map {d'} {\map f x, \map g x}$
where $f$ and $g$ are bounded mappings.
First note that we have:
{{begin-eqn}}
{{eqn | l = \size {\map f x - \map g x}
| r = \size {\map f x + \paren {-... | Let $S$ be a [[Definition:Set|set]].
Let $M = \struct {A', d'}$ be a [[Definition:Metric Space|metric space]].
Let $A$ be the [[Definition:Set|set]] of all [[Definition:Bounded Mapping to Metric Space|bounded mappings]] $f: S \to M$.
Let $d: A \times A \to \R$ be the [[Definition:Supremum Metric|supremum metric]] on... | We have that the [[Definition:Supremum Metric|supremum metric]] on $A \times A$ is defined as:
:$\ds \forall f, g \in A: \map d {f, g} := \sup_{x \mathop \in S} \map {d'} {\map f x, \map g x}$
where $f$ and $g$ are [[Definition:Bounded Mapping to Metric Space|bounded mappings]].
First note that we have:
{{begin-eqn... | Supremum Metric is Metric | https://proofwiki.org/wiki/Supremum_Metric_is_Metric | https://proofwiki.org/wiki/Supremum_Metric_is_Metric | [
"Supremum Metric"
] | [
"Definition:Set",
"Definition:Metric Space",
"Definition:Set",
"Definition:Bounded Mapping/Metric Space",
"Definition:Supremum Metric",
"Definition:Metric Space/Metric"
] | [
"Definition:Supremum Metric",
"Definition:Bounded Mapping/Metric Space",
"Triangle Inequality/Real Numbers"
] |
proofwiki-10154 | Supremum Metric on Bounded Continuous Mappings is Metric | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.
Let $A$ be the set of all continuous mappings $f: M_1 \to M_2$ which are also bounded.
Let $d: A \times A \to \R$ be the supremum metric on $A$.
Then $d$ is a metric. | The set $A$ is a subset of the set $A'$ of all bounded mappings $f: M_1 \to M_2$.
Let $d': A' \times A' \to \R$ be the supremum metric on $A'$.
From Supremum Metric is Metric, $\struct {A', d'}$ is a metric space.
By definition, $A$ is a metric subspace of $A'$.
Hence the result.
{{qed}} | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be [[Definition:Metric Space|metric spaces]].
Let $A$ be the [[Definition:Set|set]] of all [[Definition:Continuous on Metric Space|continuous mappings]] $f: M_1 \to M_2$ which are also [[Definition:Bounded Mapping to Metric Space|bounded]].
Let $d: A \time... | The set $A$ is a [[Definition:Subset|subset]] of the [[Definition:Set|set]] $A'$ of all [[Definition:Bounded Mapping to Metric Space|bounded mappings]] $f: M_1 \to M_2$.
Let $d': A' \times A' \to \R$ be the [[Definition:Supremum Metric|supremum metric]] on $A'$.
From [[Supremum Metric is Metric]], $\struct {A', d'}$ ... | Supremum Metric on Bounded Continuous Mappings is Metric | https://proofwiki.org/wiki/Supremum_Metric_on_Bounded_Continuous_Mappings_is_Metric | https://proofwiki.org/wiki/Supremum_Metric_on_Bounded_Continuous_Mappings_is_Metric | [
"Supremum Metric"
] | [
"Definition:Metric Space",
"Definition:Set",
"Definition:Continuous Mapping (Metric Space)/Space",
"Definition:Bounded Mapping/Metric Space",
"Definition:Supremum Metric",
"Definition:Metric Space/Metric"
] | [
"Definition:Subset",
"Definition:Set",
"Definition:Bounded Mapping/Metric Space",
"Definition:Supremum Metric",
"Supremum Metric is Metric",
"Definition:Metric Space",
"Definition:Metric Subspace"
] |
proofwiki-10155 | Harmonic Number is not Integer | Let $H_n$ be the $n$th harmonic number.
Then $H_n$ is not an integer for $n \ge 2$.
That is, the only harmonic numbers that are integers are $H_0$ and $H_1$. | As $H_0 = 0$ and $H_1 = 1$, they are integers.
The claim is that $H_n$ is not an integer for all $n \ge 2$.
{{AimForCont}} otherwise:
:$(\text P): \quad \exists m \in \N: H_m \in \Z$
By the definition of the harmonic numbers:
:$H_m = 1 + \dfrac 1 2 + \dfrac 1 3 + \cdots + \dfrac 1 m$
Let $2^t$ denote the highest power ... | Let $H_n$ be the $n$th [[Definition:Harmonic Number|harmonic number]].
Then $H_n$ is not an [[Definition:Integer|integer]] for $n \ge 2$.
That is, the only [[Definition:Harmonic Number|harmonic numbers]] that are [[Definition:Integer|integers]] are $H_0$ and $H_1$. | As $H_0 = 0$ and $H_1 = 1$, they are [[Definition:Integer|integers]].
The claim is that $H_n$ is not an [[Definition:Integer|integer]] for all $n \ge 2$.
{{AimForCont}} otherwise:
:$(\text P): \quad \exists m \in \N: H_m \in \Z$
By the definition of the [[Definition:Harmonic Numbers|harmonic numbers]]:
:$H_m = 1 +... | Harmonic Number is not Integer/Proof 1 | https://proofwiki.org/wiki/Harmonic_Number_is_not_Integer | https://proofwiki.org/wiki/Harmonic_Number_is_not_Integer/Proof_1 | [
"Harmonic Number is not Integer",
"Harmonic Numbers"
] | [
"Definition:Harmonic Numbers",
"Definition:Integer",
"Definition:Harmonic Numbers",
"Definition:Integer"
] | [
"Definition:Integer",
"Definition:Integer",
"Definition:Harmonic Numbers",
"Definition:Power",
"Definition:Fraction/Denominator",
"Definition:Addition/Summand",
"Definition:Set",
"Definition:Fraction/Denominator",
"Definition:Element",
"Definition:Divisor (Algebra)/Integer",
"Definition:Addition... |
proofwiki-10156 | Harmonic Number is not Integer | Let $H_n$ be the $n$th harmonic number.
Then $H_n$ is not an integer for $n \ge 2$.
That is, the only harmonic numbers that are integers are $H_0$ and $H_1$. | {{AimForCont}}:
:$(\text P): \quad \exists m \in \N: H_m \in \Z$
By the definition of the harmonic numbers:
:$1 + \dfrac 1 2 + \dfrac 1 3 + \cdots +\dfrac 1 m = H_m$
$m$ is either prime or composite.
If $m$ is prime, we have that:
{{begin-eqn}}
{{eqn | l = 1 + \frac 1 2 + \frac 1 3 + \dots + \frac 1 m
| r = H_m
}... | Let $H_n$ be the $n$th [[Definition:Harmonic Number|harmonic number]].
Then $H_n$ is not an [[Definition:Integer|integer]] for $n \ge 2$.
That is, the only [[Definition:Harmonic Number|harmonic numbers]] that are [[Definition:Integer|integers]] are $H_0$ and $H_1$. | {{AimForCont}}:
:$(\text P): \quad \exists m \in \N: H_m \in \Z$
By the definition of the [[Definition:Harmonic Numbers|harmonic numbers]]:
:$1 + \dfrac 1 2 + \dfrac 1 3 + \cdots +\dfrac 1 m = H_m$
$m$ is either [[Definition:Prime Number|prime]] or [[Definition:Composite Number|composite]].
If $m$ is [[Definition:... | Harmonic Number is not Integer/Proof 2 | https://proofwiki.org/wiki/Harmonic_Number_is_not_Integer | https://proofwiki.org/wiki/Harmonic_Number_is_not_Integer/Proof_2 | [
"Harmonic Number is not Integer",
"Harmonic Numbers"
] | [
"Definition:Harmonic Numbers",
"Definition:Integer",
"Definition:Harmonic Numbers",
"Definition:Integer"
] | [
"Definition:Harmonic Numbers",
"Definition:Prime Number",
"Definition:Composite Number",
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer",
"Definition:Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
... |
proofwiki-10157 | Supremum Metric on Differentiability Class is Metric | Let $\closedint a b \subseteq \R$ be a closed real interval.
Let $r \in \N$ be a natural number.
Let $A := \mathscr D^r \closedint a b$ be the set of all continuous functions $f: \closedint a b \to \R$ which are of differentiability class $r$.
Let $d: A \times A \to \R$ be the supremum metric on $A$.
Then $d$ is a metr... | We have that the supremum metric on $A \times A$ is defined as:
:$\ds \forall f, g \in A: \map d {f, g} := \sup_{\substack {x \mathop \in \closedint a b \\ i \mathop \in \set {0, 1, 2, \ldots, r} } } \size {\map {f^{\paren i} } x - \map {g^{\paren i} } x}$
where $f$ and $g$ are continuous functions on $\closedint a b$ ... | Let $\closedint a b \subseteq \R$ be a [[Definition:Closed Real Interval|closed real interval]].
Let $r \in \N$ be a [[Definition:Natural Number|natural number]].
Let $A := \mathscr D^r \closedint a b$ be the [[Definition:Set|set]] of all [[Definition:Continuous Real Function on Closed Interval|continuous functions]]... | We have that the [[Definition:Supremum Metric on Differentiability Class|supremum metric]] on $A \times A$ is defined as:
:$\ds \forall f, g \in A: \map d {f, g} := \sup_{\substack {x \mathop \in \closedint a b \\ i \mathop \in \set {0, 1, 2, \ldots, r} } } \size {\map {f^{\paren i} } x - \map {g^{\paren i} } x}$
whe... | Supremum Metric on Differentiability Class is Metric | https://proofwiki.org/wiki/Supremum_Metric_on_Differentiability_Class_is_Metric | https://proofwiki.org/wiki/Supremum_Metric_on_Differentiability_Class_is_Metric | [
"Supremum Metric"
] | [
"Definition:Real Interval/Closed",
"Definition:Natural Numbers",
"Definition:Set",
"Definition:Continuous Real Function/Closed Interval",
"Definition:Differentiability Class",
"Definition:Supremum Metric/Differentiability Class",
"Definition:Metric Space/Metric"
] | [
"Definition:Supremum Metric/Differentiability Class",
"Definition:Continuous Real Function/Closed Interval",
"Definition:Differentiability Class",
"Triangle Inequality/Real Numbers",
"Definition:Upper Bound of Mapping/Real-Valued",
"Definition:Supremum of Mapping/Real-Valued Function",
"Definition:Absol... |
proofwiki-10158 | Supremum Metric on Continuous Real Functions is Metric | Let $\closedint a b \subseteq \R$ be a closed real interval.
Let $\mathscr C \closedint a b$ be the set of all continuous functions $f: \closedint a b \to \R$.
Let $d$ be the supremum metric on $\mathscr C \closedint a b$.
Then $d$ is a metric. | Let $\map {\mathscr B} {\closedint a b, \R}$ be the set of all bounded real functions $f: \closedint a b \to \R$.
From Supremum Metric on Continuous Real Functions is Subspace of Bounded, $\struct {\mathscr C \closedint a b, d_{\mathscr C} }$ is a (metric) subspace of $\struct {\map {\mathscr B} {\closedint a b, \R}, d... | Let $\closedint a b \subseteq \R$ be a [[Definition:Closed Real Interval|closed real interval]].
Let $\mathscr C \closedint a b$ be the [[Definition:Set|set]] of all [[Definition:Continuous Real Function on Closed Interval|continuous functions]] $f: \closedint a b \to \R$.
Let $d$ be the [[Definition:Supremum Metric... | Let $\map {\mathscr B} {\closedint a b, \R}$ be the [[Definition:Set|set]] of all [[Definition:Bounded Real-Valued Function|bounded]] [[Definition:Real Function|real functions]] $f: \closedint a b \to \R$.
From [[Supremum Metric on Continuous Real Functions is Subspace of Bounded]], $\struct {\mathscr C \closedint a b... | Supremum Metric on Continuous Real Functions is Metric/Proof 1 | https://proofwiki.org/wiki/Supremum_Metric_on_Continuous_Real_Functions_is_Metric | https://proofwiki.org/wiki/Supremum_Metric_on_Continuous_Real_Functions_is_Metric/Proof_1 | [
"Supremum Metric",
"Supremum Metric on Continuous Real Functions is Metric"
] | [
"Definition:Real Interval/Closed",
"Definition:Set",
"Definition:Continuous Real Function/Closed Interval",
"Definition:Supremum Metric/Continuous Real Functions",
"Definition:Metric Space/Metric"
] | [
"Definition:Set",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Real Function",
"Supremum Metric on Continuous Real Functions is Subspace of Bounded",
"Definition:Metric Subspace",
"Subspace of Metric Space is Metric Space"
] |
proofwiki-10159 | Supremum Metric on Continuous Real Functions is Metric | Let $\closedint a b \subseteq \R$ be a closed real interval.
Let $\mathscr C \closedint a b$ be the set of all continuous functions $f: \closedint a b \to \R$.
Let $d$ be the supremum metric on $\mathscr C \closedint a b$.
Then $d$ is a metric. | Let $A := \mathscr D^r \closedint a b$ be the set of all continuous functions $f: \closedint a b \to \R$ which are of differentiability class $r$.
Let $d_r: A \times A \to \R$ be the supremum metric on $A$, defined as:
:$\ds \forall f, g \in A: \map d {f, g} := \sup_{\substack {x \mathop \in \closedint a b \\ i \mathop... | Let $\closedint a b \subseteq \R$ be a [[Definition:Closed Real Interval|closed real interval]].
Let $\mathscr C \closedint a b$ be the [[Definition:Set|set]] of all [[Definition:Continuous Real Function on Closed Interval|continuous functions]] $f: \closedint a b \to \R$.
Let $d$ be the [[Definition:Supremum Metric... | Let $A := \mathscr D^r \closedint a b$ be the [[Definition:Set|set]] of all [[Definition:Continuous Real Function on Closed Interval|continuous functions]] $f: \closedint a b \to \R$ which are of [[Definition:Differentiability Class|differentiability class $r$]].
Let $d_r: A \times A \to \R$ be the [[Definition:Suprem... | Supremum Metric on Continuous Real Functions is Metric/Proof 2 | https://proofwiki.org/wiki/Supremum_Metric_on_Continuous_Real_Functions_is_Metric | https://proofwiki.org/wiki/Supremum_Metric_on_Continuous_Real_Functions_is_Metric/Proof_2 | [
"Supremum Metric",
"Supremum Metric on Continuous Real Functions is Metric"
] | [
"Definition:Real Interval/Closed",
"Definition:Set",
"Definition:Continuous Real Function/Closed Interval",
"Definition:Supremum Metric/Continuous Real Functions",
"Definition:Metric Space/Metric"
] | [
"Definition:Set",
"Definition:Continuous Real Function/Closed Interval",
"Definition:Differentiability Class",
"Definition:Supremum Metric/Differentiability Class",
"Definition:Continuous Real Function/Closed Interval",
"Definition:Differentiability Class",
"Definition:Natural Numbers",
"Supremum Metr... |
proofwiki-10160 | Subspace of Metric Space is Metric Space | Let $M = \struct {A, d}$ be a metric space.
Let $H \subseteq A$.
Let $d_H: H \times H \to \R$ be the restriction $d \restriction_{H \times H}$ of $d$ to $H$.
Let $\struct {H, d_H}$ be a metric subspace of $\struct {A, d}$.
Then $d_H$ is a metric on $H$. | By definition of restriction:
:$\forall x, y \in H: \map {d_H} {x, y} = \map d {x, y}$
As $d$ is a metric, the metric space axioms are all fulfilled by all $x, y \in A$ under $d$.
As $H \subseteq A$, by definition of subset, all $x, y \in H$ are also elements of $A$.
Therefore the metric space axioms are all fulfilled ... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $H \subseteq A$.
Let $d_H: H \times H \to \R$ be the [[Definition:Restriction of Mapping|restriction]] $d \restriction_{H \times H}$ of $d$ to $H$.
Let $\struct {H, d_H}$ be a [[Definition:Metric Subspace|metric subspace]] of $\struct {A, d}... | By definition of [[Definition:Restriction of Mapping|restriction]]:
:$\forall x, y \in H: \map {d_H} {x, y} = \map d {x, y}$
As $d$ is a [[Definition:Metric|metric]], the [[Axiom:Metric Space Axioms|metric space axioms]] are all fulfilled by all $x, y \in A$ under $d$.
As $H \subseteq A$, by definition of [[Definitio... | Subspace of Metric Space is Metric Space | https://proofwiki.org/wiki/Subspace_of_Metric_Space_is_Metric_Space | https://proofwiki.org/wiki/Subspace_of_Metric_Space_is_Metric_Space | [
"Metric Subspaces"
] | [
"Definition:Metric Space",
"Definition:Restriction/Mapping",
"Definition:Metric Subspace",
"Definition:Metric Space/Metric"
] | [
"Definition:Restriction/Mapping",
"Definition:Metric Space/Metric",
"Axiom:Metric Space Axioms",
"Definition:Subset",
"Definition:Element",
"Axiom:Metric Space Axioms"
] |
proofwiki-10161 | Lp Metric on Closed Real Interval is Metric | Let $S$ be the set of all real functions which are continuous on the closed interval $\closedint a b$.
Let $p \in \R_{\ge 1}$.
Let $d_p: S \times S \to \R$ be the $L^p$ metric on $\closedint a b$:
:$\ds \forall f, g \in S: \map {d_p} {f, g} := \paren {\int_a^b \size {\map f t - \map g t}^p \rd t}^{\frac 1 p}$
Then $d_p... | === Proof of {{Metric-space-axiom|1|nolink}} ===
{{begin-eqn}}
{{eqn | l = \map {d_p} {f, f}
| r = \paren {\int_a^b \size {\map f t - \map f t}^p \rd t}^{\frac 1 p}
| c = Definition of $d_p$
}}
{{eqn | r = \paren {\int_a^b 0^p \rd t}^{\frac 1 p}
| c = {{Defof|Absolute Value}}
}}
{{eqn | r = 0
| ... | Let $S$ be the [[Definition:Set|set]] of all [[Definition:Real Function|real functions]] which are [[Definition:Continuous Real Function|continuous]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$.
Let $p \in \R_{\ge 1}$.
Let $d_p: S \times S \to \R$ be the [[Definition:Lp Metric|$L^p$ me... | === Proof of {{Metric-space-axiom|1|nolink}} ===
{{begin-eqn}}
{{eqn | l = \map {d_p} {f, f}
| r = \paren {\int_a^b \size {\map f t - \map f t}^p \rd t}^{\frac 1 p}
| c = Definition of $d_p$
}}
{{eqn | r = \paren {\int_a^b 0^p \rd t}^{\frac 1 p}
| c = {{Defof|Absolute Value}}
}}
{{eqn | r = 0
|... | Lp Metric on Closed Real Interval is Metric | https://proofwiki.org/wiki/Lp_Metric_on_Closed_Real_Interval_is_Metric | https://proofwiki.org/wiki/Lp_Metric_on_Closed_Real_Interval_is_Metric | [
"Lp Metrics"
] | [
"Definition:Set",
"Definition:Real Function",
"Definition:Continuous Real Function",
"Definition:Real Interval/Closed",
"Definition:Lp Metric",
"Definition:Metric Space/Metric"
] | [
"Integral of Constant/Definite"
] |
proofwiki-10162 | P-Sequence Space of Real Sequences is Metric Space | Let $A$ be the set of all real sequences $\sequence {x_i}$ such that the series $\ds \sum_{i \mathop \ge 0} x_i^2$ is convergent.
Let $d_p$ be the $p$-sequence metric on $\R$.
Then $\ell^p := \struct {A, d_p}$ is a metric space. | By definition of the $p$-sequence metric on $\R$:
Let $A$ be the set of all real sequences $\sequence {x_i}$ such that the series $\ds \sum_{i \mathop \ge 0} x_i^2$ is convergent.
Then $\ell^p := \struct {A, d_2}$ where $d_p: A \times A: \to \R$ is the real-valued function defined as:
:$\ds \forall x = \sequence {x_i},... | Let $A$ be the [[Definition:Set|set]] of all [[Definition:Real Sequence|real sequences]] $\sequence {x_i}$ such that the [[Definition:Series of Numbers|series]] $\ds \sum_{i \mathop \ge 0} x_i^2$ is [[Definition:Convergent Series of Numbers|convergent]].
Let $d_p$ be the [[Definition:P-Sequence Metric on Real Sequence... | By definition of the [[Definition:P-Sequence Metric on Real Sequences|$p$-sequence metric on $\R$]]:
Let $A$ be the [[Definition:Set|set]] of all [[Definition:Real Sequence|real sequences]] $\sequence {x_i}$ such that the [[Definition:Series of Numbers|series]] $\ds \sum_{i \mathop \ge 0} x_i^2$ is [[Definition:Conver... | P-Sequence Space of Real Sequences is Metric Space | https://proofwiki.org/wiki/P-Sequence_Space_of_Real_Sequences_is_Metric_Space | https://proofwiki.org/wiki/P-Sequence_Space_of_Real_Sequences_is_Metric_Space | [
"P-Sequence Metrics"
] | [
"Definition:Set",
"Definition:Real Sequence",
"Definition:Series/Number Field",
"Definition:Convergent Series/Number Field",
"Definition:P-Sequence Metric/Real Sequences",
"Definition:Metric Space"
] | [
"Definition:P-Sequence Metric/Real Sequences",
"Definition:Set",
"Definition:Real Sequence",
"Definition:Series/Number Field",
"Definition:Convergent Series/Number Field",
"Definition:Real-Valued Function"
] |
proofwiki-10163 | Peano Structure is Unique | Let $\struct {P, s, 0}$ and $\struct {P', s', 0'}$ be Peano structures.
Then there is a unique bijection $f: P \to P'$ such that:
{{begin-eqn}}
{{eqn | l = \map f 0
| r = 0'
}}
{{eqn | q = \forall n \in P
| l = \map f {\map s n}
| r = \map {s'} {\map f n}
}}
{{end-eqn}} | First to establish uniqueness of $f$.
Suppose that $f, g: P \to P'$ both satisfy the conditions.
Define $A \subseteq P$ as:
:$A := \set {n \in P: \map f n = \map g n}$
Then the first condition implies that $0 \in A$.
Now suppose that $n \in A$. Then:
{{begin-eqn}}
{{eqn| l = \map f {\map s n}
| r = \map {s'} {\map... | Let $\struct {P, s, 0}$ and $\struct {P', s', 0'}$ be [[Definition:Peano Structure|Peano structures]].
Then there is a [[Definition:Unique|unique]] [[Definition:Bijection|bijection]] $f: P \to P'$ such that:
{{begin-eqn}}
{{eqn | l = \map f 0
| r = 0'
}}
{{eqn | q = \forall n \in P
| l = \map f {\map s n... | First to establish [[Definition:Unique|uniqueness]] of $f$.
Suppose that $f, g: P \to P'$ both satisfy the conditions.
Define $A \subseteq P$ as:
:$A := \set {n \in P: \map f n = \map g n}$
Then the first condition implies that $0 \in A$.
Now suppose that $n \in A$. Then:
{{begin-eqn}}
{{eqn| l = \map f {\map s ... | Peano Structure is Unique | https://proofwiki.org/wiki/Peano_Structure_is_Unique | https://proofwiki.org/wiki/Peano_Structure_is_Unique | [
"Abstract Algebra"
] | [
"Definition:Peano Structure",
"Definition:Unique",
"Definition:Bijection"
] | [
"Definition:Unique",
"Definition:Peano Structure",
"Equality of Mappings",
"Principle of Recursive Definition/Proof 1",
"Definition:Bijection",
"Axiom:Peano's Axioms",
"Axiom:Peano's Axioms",
"Definition:Injection",
"Axiom:Peano's Axioms",
"Definition:Injection",
"Axiom:Peano's Axioms",
"Defin... |
proofwiki-10164 | P-adic Metric is Metric | Let $p \in \N$ be a prime.
Let $\norm {\,\cdot\,}_p: \Q \to \R_{\ge 0}$ be the $p$-adic norm on $\Q$.
Let $d_p$ be the $p$-adic metric on $\Q$:
:$\forall x, y \in \Q: \map {d_p} {x, y} = \norm{x - y}_p$
Then $d_p$ is a metric. | The $p$-adic metric on $\Q$ is defined as the metric induced by the $p$-adic norm on $\Q$.
It follows from Metric Induced by Norm is Metric that $d_p$ is a metric.
{{qed}}
Category:P-adic Metrics
rtedwfl78zmjc7ub1o34icmintipy67 | Let $p \in \N$ be a [[Definition:Prime Number|prime]].
Let $\norm {\,\cdot\,}_p: \Q \to \R_{\ge 0}$ be the [[Definition:P-adic Norm|$p$-adic norm]] on $\Q$.
Let $d_p$ be the [[Definition:P-adic Metric|$p$-adic metric]] on $\Q$:
:$\forall x, y \in \Q: \map {d_p} {x, y} = \norm{x - y}_p$
Then $d_p$ is a [[Definitio... | The [[Definition:P-adic Metric|$p$-adic metric]] on $\Q$ is defined as the [[Definition:Metric Induced by Norm|metric induced]] by the [[Definition:P-adic Norm|$p$-adic norm]] on $\Q$.
It follows from [[Metric Induced by Norm is Metric]] that $d_p$ is a [[Definition:Metric|metric]].
{{qed}}
[[Category:P-adic Metrics]... | P-adic Metric is Metric | https://proofwiki.org/wiki/P-adic_Metric_is_Metric | https://proofwiki.org/wiki/P-adic_Metric_is_Metric | [
"P-adic Metrics"
] | [
"Definition:Prime Number",
"Definition:P-adic Norm",
"Definition:P-adic Metric",
"Definition:Metric Space/Metric"
] | [
"Definition:P-adic Metric",
"Definition:Metric Induced by Norm",
"Definition:P-adic Norm",
"Metric Induced by Norm is Metric",
"Definition:Metric Space/Metric",
"Category:P-adic Metrics"
] |
proofwiki-10165 | Restricted P-adic Metric is Metric | Let $p \in \N$ be a prime.
Let $d^\Z_p$ be the $p$-adic metric on $\Z$:
:$\forall x, y \in \Z: \map {d^\Z_p} {x, y} = \norm {x - y}_p$
where $\norm {x - y}_p$ denotes the $p$-adic norm.
Then $d^\Z_p$ is a metric. | From $p$-adic Metric is Metric, the $p$-adic metric on $\Q$:
:$\forall x, y \in \Q: \map {d_p} {x, y} = \norm {x - y}_p$
forms a metric space $\struct {\Q, d_p}$.
The mapping:
:$\forall x, y \in \Z: \map {d^\Z_p} {x, y} = \norm {x - y}_p$
is the restriction of $d_p$ to the integers.
Hence the $p$-adic metric on $\Z$ is... | Let $p \in \N$ be a [[Definition:Prime Number|prime]].
Let $d^\Z_p$ be the [[Definition:P-adic Metric|$p$-adic metric]] on $\Z$:
:$\forall x, y \in \Z: \map {d^\Z_p} {x, y} = \norm {x - y}_p$
where $\norm {x - y}_p$ denotes the [[Definition:P-adic Norm|$p$-adic norm]].
Then $d^\Z_p$ is a [[Definition:Metric|metric... | From [[P-adic Metric is Metric|$p$-adic Metric is Metric]], the [[Definition:P-adic Metric|$p$-adic metric]] on $\Q$:
:$\forall x, y \in \Q: \map {d_p} {x, y} = \norm {x - y}_p$
forms a [[Definition:Metric Space|metric space]] $\struct {\Q, d_p}$.
The [[Definition:Mapping|mapping]]:
:$\forall x, y \in \Z: \map {d^\Z_p... | Restricted P-adic Metric is Metric | https://proofwiki.org/wiki/Restricted_P-adic_Metric_is_Metric | https://proofwiki.org/wiki/Restricted_P-adic_Metric_is_Metric | [
"P-adic Metrics"
] | [
"Definition:Prime Number",
"Definition:P-adic Metric",
"Definition:P-adic Norm",
"Definition:Metric Space/Metric"
] | [
"P-adic Metric is Metric",
"Definition:P-adic Metric",
"Definition:Metric Space",
"Definition:Mapping",
"Definition:Restriction/Mapping",
"Definition:Integer",
"Definition:P-adic Metric",
"Definition:Metric Subspace",
"Subspace of Metric Space is Metric Space"
] |
proofwiki-10166 | Constant Function is Continuous/Metric Space/Proof 2 | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.
Let $f_c: A_1 \to A_2$ be the constant mapping from $A_1$ to $A_2$:
:$\exists c \in A_2: \forall a \in A_1: \map {f_c} a = c$
That is, every point in $A_1$ maps to the same point $c$ in $A_2$.
Then $f_c$ is continuous throughout $A_1$ with ... | Let $f_c: A_1 \to A_2$ be the constant mapping between two metric spaces $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$.
Let $\epsilon \in \R_{>0}$.
Let $x \in A_1$.
Pick any $\delta \in \R_{>0}$.
Let $y \in A_1$ such that $\map {d_1} {x, y} < \delta$.
Now we have:
:$\map {f_c} x = c = \map {f_c} y$
Hence:
:... | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be [[Definition:Metric Space|metric spaces]].
Let $f_c: A_1 \to A_2$ be the [[Definition:Constant Mapping|constant mapping]] from $A_1$ to $A_2$:
:$\exists c \in A_2: \forall a \in A_1: \map {f_c} a = c$
That is, every [[Definition:Element|point]] in $A_1... | Let $f_c: A_1 \to A_2$ be the [[Definition:Constant Mapping|constant mapping]] between two [[Definition:Metric Space|metric spaces]] $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$.
Let $\epsilon \in \R_{>0}$.
Let $x \in A_1$.
Pick any $\delta \in \R_{>0}$.
Let $y \in A_1$ such that $\map {d_1} {x, y} < \... | Constant Function is Continuous/Metric Space/Proof 2 | https://proofwiki.org/wiki/Constant_Function_is_Continuous/Metric_Space/Proof_2 | https://proofwiki.org/wiki/Constant_Function_is_Continuous/Metric_Space/Proof_2 | [
"Constant Mappings",
"Continuous Mappings on Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Constant Mapping",
"Definition:Element",
"Definition:Element",
"Definition:Continuous Mapping (Metric Space)"
] | [
"Definition:Constant Mapping",
"Definition:Metric Space",
"Definition:Metric Space/Metric",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Continuous Mapping (Metric Space)"
] |
proofwiki-10167 | Constant Function is Continuous/Metric Space/Proof 1 | Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces.
Let $f_c: A_1 \to A_2$ be the constant mapping from $A_1$ to $A_2$:
:$\exists c \in A_2: \forall a \in A_1: f_c \left({a}\right) = c$
That is, every point in $A_1$ maps to the same point $c$ in $A_2$.
Then $f_c$ is continuous thro... | Let $f_c: A_1 \to A_2$ be the constant mapping between two metric spaces $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$.
From Constant Function is Uniformly Continuous, $f_c$ is uniformly continuous throughout $A_1$ with respect to $d_1$ and $d_2$.
The result follows from Uniformly Continuous Funct... | Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be [[Definition:Metric Space|metric spaces]].
Let $f_c: A_1 \to A_2$ be the [[Definition:Constant Mapping|constant mapping]] from $A_1$ to $A_2$:
:$\exists c \in A_2: \forall a \in A_1: f_c \left({a}\right) = c$
That is, every [[Definition:Eleme... | Let $f_c: A_1 \to A_2$ be the [[Definition:Constant Mapping|constant mapping]] between two [[Definition:Metric Space|metric spaces]] $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$.
From [[Constant Function is Uniformly Continuous/Metric Space|Constant Function is Uniformly Continuous]], $f_c$ is [... | Constant Function is Continuous/Metric Space/Proof 1 | https://proofwiki.org/wiki/Constant_Function_is_Continuous/Metric_Space/Proof_1 | https://proofwiki.org/wiki/Constant_Function_is_Continuous/Metric_Space/Proof_1 | [
"Constant Mappings",
"Continuous Mappings on Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Constant Mapping",
"Definition:Element",
"Definition:Element",
"Definition:Continuous Mapping (Metric Space)"
] | [
"Definition:Constant Mapping",
"Definition:Metric Space",
"Constant Function is Uniformly Continuous/Metric Space",
"Definition:Uniform Continuity/Metric Space",
"Uniformly Continuous Function is Continuous/Metric Space"
] |
proofwiki-10168 | Constant Function is Continuous/Metric Space | Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces.
Let $f_c: A_1 \to A_2$ be the constant mapping from $A_1$ to $A_2$:
:$\exists c \in A_2: \forall a \in A_1: f_c \left({a}\right) = c$
That is, every point in $A_1$ maps to the same point $c$ in $A_2$.
Then $f_c$ is continuous thro... | Let $f_c: A_1 \to A_2$ be the constant mapping between two metric spaces $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$.
From Constant Function is Uniformly Continuous, $f_c$ is uniformly continuous throughout $A_1$ with respect to $d_1$ and $d_2$.
The result follows from Uniformly Continuous Funct... | Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be [[Definition:Metric Space|metric spaces]].
Let $f_c: A_1 \to A_2$ be the [[Definition:Constant Mapping|constant mapping]] from $A_1$ to $A_2$:
:$\exists c \in A_2: \forall a \in A_1: f_c \left({a}\right) = c$
That is, every [[Definition:Eleme... | Let $f_c: A_1 \to A_2$ be the [[Definition:Constant Mapping|constant mapping]] between two [[Definition:Metric Space|metric spaces]] $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$.
From [[Constant Function is Uniformly Continuous/Metric Space|Constant Function is Uniformly Continuous]], $f_c$ is [... | Constant Function is Continuous/Metric Space/Proof 1 | https://proofwiki.org/wiki/Constant_Function_is_Continuous/Metric_Space | https://proofwiki.org/wiki/Constant_Function_is_Continuous/Metric_Space/Proof_1 | [
"Constant Mappings",
"Continuous Mappings on Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Constant Mapping",
"Definition:Element",
"Definition:Element",
"Definition:Continuous Mapping (Metric Space)"
] | [
"Definition:Constant Mapping",
"Definition:Metric Space",
"Constant Function is Uniformly Continuous/Metric Space",
"Definition:Uniform Continuity/Metric Space",
"Uniformly Continuous Function is Continuous/Metric Space"
] |
proofwiki-10169 | Constant Function is Continuous/Metric Space | Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces.
Let $f_c: A_1 \to A_2$ be the constant mapping from $A_1$ to $A_2$:
:$\exists c \in A_2: \forall a \in A_1: f_c \left({a}\right) = c$
That is, every point in $A_1$ maps to the same point $c$ in $A_2$.
Then $f_c$ is continuous thro... | Let $f_c: A_1 \to A_2$ be the constant mapping between two metric spaces $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$.
Let $\epsilon \in \R_{>0}$.
Let $x \in A_1$.
Pick any $\delta \in \R_{>0}$.
Let $y \in A_1$ such that $\map {d_1} {x, y} < \delta$.
Now we have:
:$\map {f_c} x = c = \map {f_c} y$
Hence:
:... | Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be [[Definition:Metric Space|metric spaces]].
Let $f_c: A_1 \to A_2$ be the [[Definition:Constant Mapping|constant mapping]] from $A_1$ to $A_2$:
:$\exists c \in A_2: \forall a \in A_1: f_c \left({a}\right) = c$
That is, every [[Definition:Eleme... | Let $f_c: A_1 \to A_2$ be the [[Definition:Constant Mapping|constant mapping]] between two [[Definition:Metric Space|metric spaces]] $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$.
Let $\epsilon \in \R_{>0}$.
Let $x \in A_1$.
Pick any $\delta \in \R_{>0}$.
Let $y \in A_1$ such that $\map {d_1} {x, y} < \... | Constant Function is Continuous/Metric Space/Proof 2 | https://proofwiki.org/wiki/Constant_Function_is_Continuous/Metric_Space | https://proofwiki.org/wiki/Constant_Function_is_Continuous/Metric_Space/Proof_2 | [
"Constant Mappings",
"Continuous Mappings on Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Constant Mapping",
"Definition:Element",
"Definition:Element",
"Definition:Continuous Mapping (Metric Space)"
] | [
"Definition:Constant Mapping",
"Definition:Metric Space",
"Definition:Metric Space/Metric",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Continuous Mapping (Metric Space)"
] |
proofwiki-10170 | Constant Function is Continuous/Real Function | Let $f_c: \R \to \R$ be the constant mapping:
:$\exists c \in \R: \forall a \in \R: \map {f_c} a = c$
Then $f_c$ is continuous on $\R$. | Follows directly from:
:Constant Real Function is Uniformly Continuous
:Uniformly Continuous Real Function is Continuous.
{{qed}} | Let $f_c: \R \to \R$ be the [[Definition:Constant Mapping|constant mapping]]:
:$\exists c \in \R: \forall a \in \R: \map {f_c} a = c$
Then $f_c$ is [[Definition:Continuous Real Function|continuous on $\R$]]. | Follows directly from:
:[[Constant Function is Uniformly Continuous/Real Function|Constant Real Function is Uniformly Continuous]]
:[[Uniformly Continuous Real Function is Continuous]].
{{qed}} | Constant Function is Continuous/Real Function | https://proofwiki.org/wiki/Constant_Function_is_Continuous/Real_Function | https://proofwiki.org/wiki/Constant_Function_is_Continuous/Real_Function | [
"Constant Mappings",
"Continuous Real Functions"
] | [
"Definition:Constant Mapping",
"Definition:Continuous Real Function"
] | [
"Constant Function is Uniformly Continuous/Real Function",
"Uniformly Continuous Function is Continuous/Real Function"
] |
proofwiki-10171 | Identity Mapping is Continuous/Metric Space | Let $M = \struct {A, d}$ be a metric space.
The identity mapping $I_A: A \to A$ defined as:
:$\forall x \in A: \map {I_A} x = x$
is a continuous mapping. | Let $a \in A$.
Let $\epsilon \in \R_{>0}$.
Let $\delta = \epsilon$.
Then:
{{begin-eqn}}
{{eqn | l = \map d {x, a}
| o = <
| r = \delta
| c =
}}
{{eqn | ll= \leadsto
| l = \map d {\map {I_A} x, \map {I_A} a}
| r = \map d {x, a}
| c =
}}
{{eqn | o = <
| r = \delta
| c = ... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
The [[Definition:Identity Mapping|identity mapping]] $I_A: A \to A$ defined as:
:$\forall x \in A: \map {I_A} x = x$
is a [[Definition:Continuous Mapping (Metric Spaces)|continuous mapping]]. | Let $a \in A$.
Let $\epsilon \in \R_{>0}$.
Let $\delta = \epsilon$.
Then:
{{begin-eqn}}
{{eqn | l = \map d {x, a}
| o = <
| r = \delta
| c =
}}
{{eqn | ll= \leadsto
| l = \map d {\map {I_A} x, \map {I_A} a}
| r = \map d {x, a}
| c =
}}
{{eqn | o = <
| r = \delta
| c... | Identity Mapping is Continuous/Metric Space | https://proofwiki.org/wiki/Identity_Mapping_is_Continuous/Metric_Space | https://proofwiki.org/wiki/Identity_Mapping_is_Continuous/Metric_Space | [
"Continuous Mappings on Metric Spaces",
"Identity Mapping is Continuous"
] | [
"Definition:Metric Space",
"Definition:Identity Mapping",
"Definition:Continuous Mapping (Metric Space)"
] | [] |
proofwiki-10172 | Identity Mapping on Real Vector Space from Chebyshev to Euclidean Metric is Continuous | Let $\R^n$ be an $n$-dimensional real vector space.
Let $d_2$ be the Euclidean metric on $\R^n$.
Let $d_\infty$ be the Chebyshev distance on $\R^n$.
Let $I: \R^n \to \R^n$ be the identity mapping from $\R^n$ to itself.
Then the mapping:
:$I: \struct {\R^n, d_\infty} \to \struct {\R^n, d_2}$
is $\tuple {d_\infty, d_2}$-... | Let $a = \tuple {a_1, a_2, \ldots, a_n} \in \R^n$.
Let $\epsilon \in \R_{>0}$.
Let $\delta = \dfrac {\epsilon} {\sqrt n}$.
Let $x = \tuple {x_1, x_2, \ldots, x_n}$ be such that $\map {d_\infty} {x, a} < \delta$.
That is:
:$\ds \max_{i \mathop \le i \mathop \le n} \set {\size {a_i - x_i} } < \delta$
Then:
{{begin-eqn}}
... | Let $\R^n$ be an [[Definition:Dimension of Vector Space|$n$-dimensional]] [[Definition:Real Vector Space|real vector space]].
Let $d_2$ be the [[Definition:Euclidean Metric on Real Vector Space|Euclidean metric]] on $\R^n$.
Let $d_\infty$ be the [[Definition:Chebyshev Distance on Real Vector Space|Chebyshev distance]... | Let $a = \tuple {a_1, a_2, \ldots, a_n} \in \R^n$.
Let $\epsilon \in \R_{>0}$.
Let $\delta = \dfrac {\epsilon} {\sqrt n}$.
Let $x = \tuple {x_1, x_2, \ldots, x_n}$ be such that $\map {d_\infty} {x, a} < \delta$.
That is:
:$\ds \max_{i \mathop \le i \mathop \le n} \set {\size {a_i - x_i} } < \delta$
Then:
{{begin-... | Identity Mapping on Real Vector Space from Chebyshev to Euclidean Metric is Continuous | https://proofwiki.org/wiki/Identity_Mapping_on_Real_Vector_Space_from_Chebyshev_to_Euclidean_Metric_is_Continuous | https://proofwiki.org/wiki/Identity_Mapping_on_Real_Vector_Space_from_Chebyshev_to_Euclidean_Metric_is_Continuous | [
"Continuous Mappings on Metric Spaces",
"Identity Mappings",
"Euclidean Metric",
"Chebyshev Distance"
] | [
"Definition:Dimension of Vector Space",
"Definition:Real Vector Space",
"Definition:Euclidean Metric/Real Vector Space",
"Definition:Chebyshev Distance/Real Vector Space",
"Definition:Identity Mapping",
"Definition:Mapping",
"Definition:Continuous Mapping (Metric Space)"
] | [
"Definition:Continuous Mapping (Metric Space)"
] |
proofwiki-10173 | Identity Mapping on Real Vector Space from Euclidean to Chebyshev Distance is Continuous | Let $\R^n$ be an $n$-dimensional real vector space.
Let $d_2$ be the Euclidean metric on $\R^n$.
Let $d_\infty$ be the Chebyshev distance on $\R^n$.
Let $I: \R^n \to \R^n$ be the identity mapping from $\R^n$ to itself.
Then the mapping:
:$I: \struct {\R^n, d_2} \to \struct {\R^n, d_\infty}$
is $\tuple {d_2, d_\infty}$-... | Let $a = \tuple {a_1, a_2, \ldots, a_n} \in \R^n$.
Let $\epsilon \in \R_{>0}$.
Let $\delta = \epsilon$.
Let $x = \tuple {x_1, x_2, \ldots, x_n}$ be such that $\map {d_2} {x, a} < \delta$.
That is:
:$\ds \sqrt {\sum_{i \mathop = i}^n \paren {a_i - x_i} } < \delta$
Then:
{{begin-eqn}}
{{eqn | l = \sum_{i \mathop = i}^n \... | Let $\R^n$ be an [[Definition:Dimension of Vector Space|$n$-dimensional]] [[Definition:Real Vector Space|real vector space]].
Let $d_2$ be the [[Definition:Euclidean Metric on Real Vector Space|Euclidean metric]] on $\R^n$.
Let $d_\infty$ be the [[Definition:Chebyshev Distance on Real Vector Space|Chebyshev distance]... | Let $a = \tuple {a_1, a_2, \ldots, a_n} \in \R^n$.
Let $\epsilon \in \R_{>0}$.
Let $\delta = \epsilon$.
Let $x = \tuple {x_1, x_2, \ldots, x_n}$ be such that $\map {d_2} {x, a} < \delta$.
That is:
:$\ds \sqrt {\sum_{i \mathop = i}^n \paren {a_i - x_i} } < \delta$
Then:
{{begin-eqn}}
{{eqn | l = \sum_{i \mathop = ... | Identity Mapping on Real Vector Space from Euclidean to Chebyshev Distance is Continuous | https://proofwiki.org/wiki/Identity_Mapping_on_Real_Vector_Space_from_Euclidean_to_Chebyshev_Distance_is_Continuous | https://proofwiki.org/wiki/Identity_Mapping_on_Real_Vector_Space_from_Euclidean_to_Chebyshev_Distance_is_Continuous | [
"Continuous Mappings on Metric Spaces",
"Identity Mappings",
"Euclidean Metric",
"Chebyshev Distance"
] | [
"Definition:Dimension of Vector Space",
"Definition:Real Vector Space",
"Definition:Euclidean Metric/Real Vector Space",
"Definition:Chebyshev Distance/Real Vector Space",
"Definition:Identity Mapping",
"Definition:Mapping",
"Definition:Continuous Mapping (Metric Space)"
] | [
"Definition:Continuous Mapping (Metric Space)"
] |
proofwiki-10174 | Composite of Continuous Mappings at Point between Metric Spaces is Continuous at Point | Let $M_1 = \struct {X_1, d_1}, M_2 = \struct {X_2, d_2}, M_3 = \struct {X_3, d_3}$ be metric spaces.
Let $f: M_1 \to M_2$ be continuous at $a \in X_1$.
Let $g: M_2 \to M_3$ be continuous at $\map f a \in X_2$.
Then their composite $g \circ f: M_1 \to M_3$ is continuous at $a \in X_1$. | Let $\epsilon \in \R_{>0}$.
The strategy is to find a $\delta \in \R_{>0}$ such that:
:$\map {d_1} {x, a} < \delta \implies \map {d_3} {\map g {\map f x}, \map g {\map f a} } < \epsilon$
As $g$ is continuous at $\map f a$:
:$\exists \eta \in \R_{>0}: \forall y \in X_2: \map {d_2} {y, \map f a} < \eta \implies \map {d_3... | Let $M_1 = \struct {X_1, d_1}, M_2 = \struct {X_2, d_2}, M_3 = \struct {X_3, d_3}$ be [[Definition:Metric Space|metric spaces]].
Let $f: M_1 \to M_2$ be [[Definition:Continuous at Point of Metric Space|continuous at $a \in X_1$]].
Let $g: M_2 \to M_3$ be [[Definition:Continuous at Point of Metric Space|continuous at ... | Let $\epsilon \in \R_{>0}$.
The strategy is to find a $\delta \in \R_{>0}$ such that:
:$\map {d_1} {x, a} < \delta \implies \map {d_3} {\map g {\map f x}, \map g {\map f a} } < \epsilon$
As $g$ is [[Definition:Continuous at Point of Metric Space|continuous at $\map f a$]]:
:$\exists \eta \in \R_{>0}: \forall y \in X... | Composite of Continuous Mappings at Point between Metric Spaces is Continuous at Point | https://proofwiki.org/wiki/Composite_of_Continuous_Mappings_at_Point_between_Metric_Spaces_is_Continuous_at_Point | https://proofwiki.org/wiki/Composite_of_Continuous_Mappings_at_Point_between_Metric_Spaces_is_Continuous_at_Point | [
"Metric Spaces",
"Continuous Mappings on Metric Spaces",
"Composite Mappings"
] | [
"Definition:Metric Space",
"Definition:Continuous Mapping (Metric Space)/Point",
"Definition:Continuous Mapping (Metric Space)/Point",
"Definition:Composition of Mappings",
"Definition:Continuous Mapping (Metric Space)/Point"
] | [
"Definition:Continuous Mapping (Metric Space)/Point",
"Definition:Continuous Mapping (Metric Space)/Point"
] |
proofwiki-10175 | Composite of Continuous Mappings between Metric Spaces is Continuous | Let $M_1 = \struct {X_1, d_1}$, $M_2 = \struct {X_2, d_2}$ and $M_3 = \struct {X_3, d_3}$ be metric spaces.
Let $f: M_1 \to M_2$ and $g: M_2 \to M_3$ be continuous mappings.
Then their composite $g \circ f: M_1 \to M_3$ is continuous. | From Metric Induces Topology, the metric spaces described are topological spaces.
The result follows from Composite of Continuous Mappings is Continuous.
{{qed}} | Let $M_1 = \struct {X_1, d_1}$, $M_2 = \struct {X_2, d_2}$ and $M_3 = \struct {X_3, d_3}$ be [[Definition:Metric Space|metric spaces]].
Let $f: M_1 \to M_2$ and $g: M_2 \to M_3$ be [[Definition:Continuous Mapping (Metric Spaces)|continuous mappings]].
Then their [[Definition:Composition of Mappings|composite]] $g \c... | From [[Metric Induces Topology]], the [[Definition:Metric Space|metric spaces]] described are [[Definition:Topological Space|topological spaces]].
The result follows from [[Composite of Continuous Mappings is Continuous]].
{{qed}} | Composite of Continuous Mappings between Metric Spaces is Continuous/Proof 1 | https://proofwiki.org/wiki/Composite_of_Continuous_Mappings_between_Metric_Spaces_is_Continuous | https://proofwiki.org/wiki/Composite_of_Continuous_Mappings_between_Metric_Spaces_is_Continuous/Proof_1 | [
"Metric Spaces",
"Continuous Mappings on Metric Spaces",
"Composite Mappings",
"Composite of Continuous Mappings between Metric Spaces is Continuous"
] | [
"Definition:Metric Space",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Composition of Mappings",
"Definition:Continuous Mapping (Metric Space)"
] | [
"Metric Induces Topology",
"Definition:Metric Space",
"Definition:Topological Space",
"Composite of Continuous Mappings is Continuous"
] |
proofwiki-10176 | Composite of Continuous Mappings between Metric Spaces is Continuous | Let $M_1 = \struct {X_1, d_1}$, $M_2 = \struct {X_2, d_2}$ and $M_3 = \struct {X_3, d_3}$ be metric spaces.
Let $f: M_1 \to M_2$ and $g: M_2 \to M_3$ be continuous mappings.
Then their composite $g \circ f: M_1 \to M_3$ is continuous. | Let $f$ and $g$ be continuous mappings.
By definition:
:$f$ is continuous at $a \in X_1$ for all $a \in X_1$
:$g$ is continuous at $\map f a \in X_2$ for all $\map f a \in X_2$.
The result follows from Composite of Continuous Mappings at Point between Metric Spaces is Continuous at Point
{{qed}} | Let $M_1 = \struct {X_1, d_1}$, $M_2 = \struct {X_2, d_2}$ and $M_3 = \struct {X_3, d_3}$ be [[Definition:Metric Space|metric spaces]].
Let $f: M_1 \to M_2$ and $g: M_2 \to M_3$ be [[Definition:Continuous Mapping (Metric Spaces)|continuous mappings]].
Then their [[Definition:Composition of Mappings|composite]] $g \c... | Let $f$ and $g$ be [[Definition:Continuous Mapping (Metric Spaces)|continuous mappings]].
By definition:
:$f$ is [[Definition:Continuous at Point of Metric Space|continuous at $a \in X_1$]] for all $a \in X_1$
:$g$ is [[Definition:Continuous at Point of Metric Space|continuous at $\map f a \in X_2$]] for all $\map f a... | Composite of Continuous Mappings between Metric Spaces is Continuous/Proof 2 | https://proofwiki.org/wiki/Composite_of_Continuous_Mappings_between_Metric_Spaces_is_Continuous | https://proofwiki.org/wiki/Composite_of_Continuous_Mappings_between_Metric_Spaces_is_Continuous/Proof_2 | [
"Metric Spaces",
"Continuous Mappings on Metric Spaces",
"Composite Mappings",
"Composite of Continuous Mappings between Metric Spaces is Continuous"
] | [
"Definition:Metric Space",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Composition of Mappings",
"Definition:Continuous Mapping (Metric Space)"
] | [
"Definition:Continuous Mapping (Metric Space)",
"Definition:Continuous Mapping (Metric Space)/Point",
"Definition:Continuous Mapping (Metric Space)/Point",
"Composite of Continuous Mappings at Point between Metric Spaces is Continuous at Point"
] |
proofwiki-10177 | Metric Space Completeness is not Preserved by Homeomorphism | 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 a homeomorphism.
If $M_1$ is complete then it is not necessarily the case that so is $M_2$. | Let $\Z_{>0}$ be the set of (strictly) positive integers.
Let $d: \Z_{>0} \times \Z_{>0} \to \R$ be the usual (Euclidean) metric on $\Z_{>0}$.
Let $\delta: \Z_{>0} \times \Z_{>0} \to \R$ be the metric on $\Z_{>0}$ defined as:
:$\forall x, y \in \Z_{>0}: \map \delta {x, y} = \dfrac {\size {x - y} } {x y}$
Let $\tau_d$ d... | 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 a [[Definition:Homeomorphism (Metric Spaces)|homeomorphism]].
If $M_1$ is [[Definition:Complete Metric Space|complete]] then it is not necessarily the case that so is $M_2$. | Let $\Z_{>0}$ be the [[Definition:Set|set]] of [[Definition:Strictly Positive Integer|(strictly) positive integers]].
Let $d: \Z_{>0} \times \Z_{>0} \to \R$ be the [[Definition:Euclidean Metric on Real Number Line|usual (Euclidean) metric]] on $\Z_{>0}$.
Let $\delta: \Z_{>0} \times \Z_{>0} \to \R$ be the [[Definition... | Metric Space Completeness is not Preserved by Homeomorphism | https://proofwiki.org/wiki/Metric_Space_Completeness_is_not_Preserved_by_Homeomorphism | https://proofwiki.org/wiki/Metric_Space_Completeness_is_not_Preserved_by_Homeomorphism | [
"Complete Metric Spaces",
"Homeomorphisms (Metric Spaces)"
] | [
"Definition:Metric Space",
"Definition:Homeomorphism/Metric Spaces",
"Definition:Complete Metric Space"
] | [
"Definition:Set",
"Definition:Strictly Positive/Integer",
"Definition:Euclidean Metric/Real Number Line",
"Definition:Metric Space/Metric",
"Definition:Topology Induced by Metric",
"Definition:Topology Induced by Metric",
"Topologies induced by Usual Metric and Scaled Euclidean Metric on Positive Intege... |
proofwiki-10178 | Mapping from L1 Space to Real Number Space is Continuous | Let $\struct {\R, d}$ be the real number line under the usual metric $d$.
Let $X$ be the set of continuous real functions $f: \closedint a b \to \R$.
Let $d_1$ be the $L^1$ metric on $X$.
Let $I: X \to \R$ be the real-valued function defined as:
:$\ds \forall f \in X: \map I f := \int_a^b \map f t \ \mathop d t$
Then t... | The $L^1$ metric on $X$ is defined as:
:$\ds \forall f, g \in S: \map {d_1} {f, g} := \int_a^b \size {\map f t - \map g t} \rd t$
Let $\epsilon \in \R_{>0}$.
Let $f \in X$.
Let $\delta = \epsilon$.
Then:
{{begin-eqn}}
{{eqn | q = \forall g \in X
| l = \map {d_1} {f, g}
| o = <
| r = \delta
| c =... | Let $\struct {\R, d}$ be the [[Definition:Real Number Line|real number line]] under the [[Definition:Usual Metric|usual metric]] $d$.
Let $X$ be the [[Definition:Set|set]] of [[Definition:Continuous Real Function|continuous real functions]] $f: \closedint a b \to \R$.
Let $d_1$ be the [[Definition:L1 Metric on Closed... | The [[Definition:L1 Metric on Closed Real Interval|$L^1$ metric]] on $X$ is defined as:
:$\ds \forall f, g \in S: \map {d_1} {f, g} := \int_a^b \size {\map f t - \map g t} \rd t$
Let $\epsilon \in \R_{>0}$.
Let $f \in X$.
Let $\delta = \epsilon$.
Then:
{{begin-eqn}}
{{eqn | q = \forall g \in X
| l = \map {d_1... | Mapping from L1 Space to Real Number Space is Continuous | https://proofwiki.org/wiki/Mapping_from_L1_Space_to_Real_Number_Space_is_Continuous | https://proofwiki.org/wiki/Mapping_from_L1_Space_to_Real_Number_Space_is_Continuous | [
"Continuous Mappings",
"L1 Metric"
] | [
"Definition:Real Number/Real Number Line",
"Definition:Usual Metric",
"Definition:Set",
"Definition:Continuous Real Function",
"Definition:L1 Metric/Closed Real Interval",
"Definition:Real-Valued Function",
"Definition:Mapping",
"Definition:Continuous Mapping (Metric Space)"
] | [
"Definition:L1 Metric/Closed Real Interval",
"Triangle Inequality for Integrals/Real",
"Linear Combination of Integrals/Definite",
"Definition:Continuous Mapping (Metric Space)/Point",
"Definition:Continuous Mapping (Metric Space)/Point",
"Definition:Continuous Mapping (Metric Space)/Point",
"Definition... |
proofwiki-10179 | Cartesian Product under Chebyshev Distance of Continuous Mappings between Metric Spaces is Continuous | Let $n \in \N_{>0}$.
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 $N_1 = \struct {B_1, d'_1}, N_2 = \struct {B_2, d'_2}, \ldots, N_n = \struct {B_n, d'_n}$ be metric spaces.
Let $f_i: M_i \to N_i$ be continuous mappings for all $i \in \set {1, 2, \ldot... | Let $\epsilon \in \R_{>0}$.
Let $x \in \AA$.
Let $k \in \left\{{1, 2, \ldots, n}\right\}$.
Then as $f_k$ is continuous:
:$(1): \quad \exists \delta_k \in \R_{>0}: \forall y_k \in A_k: \map {d_k} {x_k, y_k} < \delta_k \implies \map {d'} {\map {f_k} {x_k}, \map {f_k} {y_k} } < \epsilon$
Let $\delta = \max \set {\delta_k:... | Let $n \in \N_{>0}$.
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 $N_1 = \struct {B_1, d'_1}, N_2 = \struct {B_2, d'_2}, \ldots, N_n = \struct {B_n, d'_n}$ be [[Definition:Metric Space|metric spaces]].
Let $f_i: M_i \to N... | Let $\epsilon \in \R_{>0}$.
Let $x \in \AA$.
Let $k \in \left\{{1, 2, \ldots, n}\right\}$.
Then as $f_k$ is [[Definition:Continuous Mapping (Metric Spaces)|continuous]]:
:$(1): \quad \exists \delta_k \in \R_{>0}: \forall y_k \in A_k: \map {d_k} {x_k, y_k} < \delta_k \implies \map {d'} {\map {f_k} {x_k}, \map {f_k} {... | Cartesian Product under Chebyshev Distance of Continuous Mappings between Metric Spaces is Continuous | https://proofwiki.org/wiki/Cartesian_Product_under_Chebyshev_Distance_of_Continuous_Mappings_between_Metric_Spaces_is_Continuous | https://proofwiki.org/wiki/Cartesian_Product_under_Chebyshev_Distance_of_Continuous_Mappings_between_Metric_Spaces_is_Continuous | [
"Continuous Mappings on Metric Spaces",
"Chebyshev Distance"
] | [
"Definition:Metric Space",
"Definition:Metric Space",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Cartesian Product/Finite",
"Definition:Cartesian Product/Finite",
"Definition:Chebyshev Distance/General Definition",
"Definition:Mapping",
"Definition:Continuous Mapping (Metric Space)"
] | [
"Definition:Continuous Mapping (Metric Space)",
"Definition:Continuous Mapping (Metric Space)/Point",
"Definition:Continuous Mapping (Metric Space)/Point",
"Definition:Continuous Mapping (Metric Space)/Point",
"Definition:Continuous Mapping (Metric Space)"
] |
proofwiki-10180 | Addition of Coordinates on Cartesian Plane under Chebyshev Distance is Continuous Function | Let $\R^2$ be the real number plane.
Let $d_\infty$ be the Chebyshev distance on $\R^2$.
Let $f: \R^2 \to \R$ be the real-valued function defined as:
:$\forall \tuple {x_1, x_2} \in \R^2: \map f {x_1, x_2} = x_1 + x_2$
Then $f$ is continuous. | First we note that:
{{begin-eqn}}
{{eqn | l = \size {\paren {x_1 + x_2} - \paren {y_1 + y_2} }
| r = \size {\paren {x_1 - y_1} + \paren {x_2 - y_2} }
| c =
}}
{{eqn | o = \le
| r = \size {x_1 - y_1} + \size {x_2 - y_2}
| c = Triangle Inequality for Real Numbers
}}
{{eqn | n = 1
| o = \le
... | Let $\R^2$ be the [[Definition:Real Number Plane|real number plane]].
Let $d_\infty$ be the [[Definition:Chebyshev Distance on Real Number Plane|Chebyshev distance]] on $\R^2$.
Let $f: \R^2 \to \R$ be the [[Definition:Real-Valued Function|real-valued function]] defined as:
:$\forall \tuple {x_1, x_2} \in \R^2: \map f... | First we note that:
{{begin-eqn}}
{{eqn | l = \size {\paren {x_1 + x_2} - \paren {y_1 + y_2} }
| r = \size {\paren {x_1 - y_1} + \paren {x_2 - y_2} }
| c =
}}
{{eqn | o = \le
| r = \size {x_1 - y_1} + \size {x_2 - y_2}
| c = [[Triangle Inequality for Real Numbers]]
}}
{{eqn | n = 1
| o =... | Addition of Coordinates on Cartesian Plane under Chebyshev Distance is Continuous Function | https://proofwiki.org/wiki/Addition_of_Coordinates_on_Cartesian_Plane_under_Chebyshev_Distance_is_Continuous_Function | https://proofwiki.org/wiki/Addition_of_Coordinates_on_Cartesian_Plane_under_Chebyshev_Distance_is_Continuous_Function | [
"Continuous Mappings on Metric Spaces",
"Chebyshev Distance"
] | [
"Definition:Real Number Plane",
"Definition:Chebyshev Distance/Real Number Plane",
"Definition:Real-Valued Function",
"Definition:Continuous Mapping (Metric Space)"
] | [
"Triangle Inequality/Real Numbers",
"Definition:Continuous Mapping (Metric Space)/Point",
"Definition:Continuous Mapping (Metric Space)/Point",
"Definition:Continuous Mapping (Metric Space)/Point",
"Definition:Continuous Mapping (Metric Space)"
] |
proofwiki-10181 | Addition of Coordinates on Euclidean Plane is Continuous Function | Let $\struct {\R^2, d_2}$ be the real number plane with the usual (Euclidean) metric.
Let $f: \R^2 \to \R$ be the real-valued function defined as:
:$\forall \tuple {x_1, x_2} \in \R^2: \map f {x_1, x_2} = x_1 + x_2$
Then $f$ is continuous. | First we note that:
{{begin-eqn}}
{{eqn | o =
| r = \size {\paren {x_1 + x_2} - \paren {y_1 + y_2} }
| c =
}}
{{eqn | r = \size {\paren {x_1 - y_1} + \paren {x_2 - y_2} }
| c =
}}
{{eqn | o = \le
| r = \size {x_1 - y_1} + \size {x_2 - y_2}
| c = Triangle Inequality for Real Numbers
}}
{... | Let $\struct {\R^2, d_2}$ be the [[Definition:Real Number Plane with Euclidean Metric|real number plane with the usual (Euclidean) metric]].
Let $f: \R^2 \to \R$ be the [[Definition:Real-Valued Function|real-valued function]] defined as:
:$\forall \tuple {x_1, x_2} \in \R^2: \map f {x_1, x_2} = x_1 + x_2$
Then $f$ i... | First we note that:
{{begin-eqn}}
{{eqn | o =
| r = \size {\paren {x_1 + x_2} - \paren {y_1 + y_2} }
| c =
}}
{{eqn | r = \size {\paren {x_1 - y_1} + \paren {x_2 - y_2} }
| c =
}}
{{eqn | o = \le
| r = \size {x_1 - y_1} + \size {x_2 - y_2}
| c = [[Triangle Inequality for Real Numbers]]... | Addition of Coordinates on Euclidean Plane is Continuous Function | https://proofwiki.org/wiki/Addition_of_Coordinates_on_Euclidean_Plane_is_Continuous_Function | https://proofwiki.org/wiki/Addition_of_Coordinates_on_Euclidean_Plane_is_Continuous_Function | [
"Continuous Mappings on Metric Spaces",
"Real Number Plane with Euclidean Metric"
] | [
"Definition:Euclidean Metric/Real Number Plane",
"Definition:Real-Valued Function",
"Definition:Continuous Mapping (Metric Space)"
] | [
"Triangle Inequality/Real Numbers",
"P-Product Metrics on Real Vector Space are Topologically Equivalent",
"Definition:Continuous Mapping (Metric Space)/Point",
"Definition:Continuous Mapping (Metric Space)/Point",
"Definition:Continuous Mapping (Metric Space)/Point",
"Definition:Continuous Mapping (Metri... |
proofwiki-10182 | Metric Space Continuity by Inverse of Mapping between Open Balls | 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 from $A_1$ to $A_2$.
Let $a \in A_1$ be a point in $A_1$.
$f$ is continuous at $a$ with respect to the metrics $d_1$ and $d_2$ {{iff}}:
:$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \map {B_\d... | By definition, $f$ is continuous at $a$ with respect to the metrics $d_1$ and $d_2$ {{iff}}:
:$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: f \sqbrk {\map {B_\delta} {a; d_1} } \subseteq \map {B_\epsilon} {\map f a; d_2}$
For a mapping $f: X \to Y$ we have:
:$f \sqbrk U \subseteq V \iff U \subseteq f^{-1} ... | 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]] from $A_1$ to $A_2$.
Let $a \in A_1$ be a point in $A_1$.
$f$ is [[Definition:Continuous at Point of Metric Space|continuous at $a$]] with respect to t... | By definition, $f$ is [[Definition:Continuous at Point of Metric Space|continuous at $a$]] with respect to the [[Definition:Metric|metrics]] $d_1$ and $d_2$ {{iff}}:
:$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: f \sqbrk {\map {B_\delta} {a; d_1} } \subseteq \map {B_\epsilon} {\map f a; d_2}$
For a [[De... | Metric Space Continuity by Inverse of Mapping between Open Balls | https://proofwiki.org/wiki/Metric_Space_Continuity_by_Inverse_of_Mapping_between_Open_Balls | https://proofwiki.org/wiki/Metric_Space_Continuity_by_Inverse_of_Mapping_between_Open_Balls | [
"Open Balls",
"Continuous Mappings on Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Mapping",
"Definition:Continuous Mapping (Metric Space)/Point",
"Definition:Metric Space/Metric",
"Definition:Open Ball",
"Definition:Metric Space/Metric"
] | [
"Definition:Continuous Mapping (Metric Space)/Point",
"Definition:Metric Space/Metric",
"Definition:Mapping"
] |
proofwiki-10183 | Open Ball is Neighborhood of all Points Inside | Let $M = \struct {A, d}$ be a metric space.
Let $a \in A$.
Let $\map {B_\epsilon} a$ be an open $\epsilon$-ball of $a$ in $M$.
Let $x \in \map {B_\epsilon} a$.
Then $\map {B_\epsilon} a$ is a neighborhood of $x$ in $M$. | From Open Ball of Point Inside Open Ball:
:$\exists \delta \in \R: \map {B_\delta} x \subseteq \map {B_\epsilon} a$
Thus by definition $\map {B_\epsilon} a$ is a neighborhood of $x$ in $M$.
{{qed}} | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $a \in A$.
Let $\map {B_\epsilon} a$ be an [[Definition:Open Ball|open $\epsilon$-ball]] of $a$ in $M$.
Let $x \in \map {B_\epsilon} a$.
Then $\map {B_\epsilon} a$ is a [[Definition:Neighborhood (Metric Space)|neighborhood]] of $x$ in $M$. | From [[Open Ball of Point Inside Open Ball]]:
:$\exists \delta \in \R: \map {B_\delta} x \subseteq \map {B_\epsilon} a$
Thus by definition $\map {B_\epsilon} a$ is a [[Definition:Neighborhood (Metric Space)|neighborhood]] of $x$ in $M$.
{{qed}} | Open Ball is Neighborhood of all Points Inside | https://proofwiki.org/wiki/Open_Ball_is_Neighborhood_of_all_Points_Inside | https://proofwiki.org/wiki/Open_Ball_is_Neighborhood_of_all_Points_Inside | [
"Open Balls",
"Neighborhoods"
] | [
"Definition:Metric Space",
"Definition:Open Ball",
"Definition:Neighborhood (Metric Space)"
] | [
"Open Ball of Point Inside Open Ball",
"Definition:Neighborhood (Metric Space)"
] |
proofwiki-10184 | Metric Space Continuity by Neighborhood | 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 from $A_1$ to $A_2$.
Let $a \in A_1$ be a point in $A_1$.
Then the following definitions of continuity of $f$ at $a$ with respect to $d_1$ and $d_2$ are equivalent: | === $\epsilon$-Ball Definition implies Definition by Neighborhood ===
Suppose that $f$ is $\tuple {d_1, d_2}$-continuous at $a$ in the sense that:
:$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: f \sqbrk {\map {B_\delta} {a; d_1} } \subseteq \map {B_\epsilon} {\map f a; d_2}$
where $\map {B_\epsilon} {\map ... | 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]] from $A_1$ to $A_2$.
Let $a \in A_1$ be a point in $A_1$.
Then the following definitions of [[Definition:Continuous at Point of Metric Space|continuity... | === $\epsilon$-Ball Definition implies Definition by Neighborhood ===
Suppose that [[Definition:Continuous Mapping (Metric Space)/Point/Definition 3|$f$ is $\tuple {d_1, d_2}$-continuous at $a$]] in the sense that:
:$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: f \sqbrk {\map {B_\delta} {a; d_1} } \subset... | Metric Space Continuity by Neighborhood | https://proofwiki.org/wiki/Metric_Space_Continuity_by_Neighborhood | https://proofwiki.org/wiki/Metric_Space_Continuity_by_Neighborhood | [
"Neighborhoods",
"Continuous Mappings on Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Mapping",
"Definition:Continuous Mapping (Metric Space)/Point",
"Definition:Logical Equivalence"
] | [
"Definition:Continuous Mapping (Metric Space)/Point/Definition 3",
"Definition:Open Ball",
"Definition:Metric Space/Metric",
"Definition:Neighborhood (Metric Space)",
"Definition:Continuous Mapping (Metric Space)/Point/Definition 3",
"Open Ball is Neighborhood of all Points Inside",
"Definition:Neighbor... |
proofwiki-10185 | Sequence Characterization of Open Sets | Let $\struct {X, d}$ be a metric space.
Let $G \subseteq X$.
{{TFAE}}
{{begin-itemize}}
{{item|(1):|$G \subseteq X$ is an open set of $\struct {X, d}$}}
{{item|(2):|$\forall x \in G: \forall \sequence {x_n} \in X: x_n \to x: \exists n_0 \in \N: \forall n \ge n_0: \sequence {x_n} \in G$}}
{{end-itemize}} | === $(1)$ implies $(2)$ ===
Let $G \subseteq X$ be open in $\struct {X, d}$.
Let $x \in G$.
Let $\sequence {x_n}$ be a sequence in $X$ such that $x_n \to x$.
By definition of open set, there exists $\epsilon > 0$ such that:
: $B_\epsilon \left({x}\right) \subseteq G$
where $B_\epsilon \left({x}\right)$ is the open $\ep... | Let $\struct {X, d}$ be a [[Definition:Metric Space|metric space]].
Let $G \subseteq X$.
{{TFAE}}
{{begin-itemize}}
{{item|(1):|$G \subseteq X$ is an [[Definition:Open Set (Metric Space)|open set]] of $\struct {X, d}$}}
{{item|(2):|$\forall x \in G: \forall \sequence {x_n} \in X: x_n \to x: \exists n_0 \in \N: \fora... | === $(1)$ implies $(2)$ ===
Let $G \subseteq X$ be [[Definition:Open Set (Metric Space)|open]] in $\struct {X, d}$.
Let $x \in G$.
Let $\sequence {x_n}$ be a [[Definition:Sequence|sequence]] in $X$ such that $x_n \to x$.
By definition of [[Definition:Open Set (Metric Space)|open set]], there exists $\epsilon > 0$ s... | Sequence Characterization of Open Sets | https://proofwiki.org/wiki/Sequence_Characterization_of_Open_Sets | https://proofwiki.org/wiki/Sequence_Characterization_of_Open_Sets | [
"Open Sets"
] | [
"Definition:Metric Space",
"Definition:Open Set/Metric Space"
] | [
"Definition:Open Set/Metric Space",
"Definition:Sequence",
"Definition:Open Set/Metric Space",
"Definition:Open Ball",
"Definition:Open Set/Metric Space",
"Definition:Sequence",
"Definition:Open Set/Metric Space"
] |
proofwiki-10186 | Metric Space Continuity by Inverse of Mapping between Neighborhoods | 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 from $A_1$ to $A_2$.
Let $a \in A_1$ be a point in $A_1$.
$f$ is continuous at $a$ with respect to the metrics $d_1$ and $d_2$ {{iff}}:
:for each neighborhood $N$ of $\map f a$ in $M_2$, $f^{-1} \sqbrk N$ i... | By definition, $f$ is continuous at $a$ with respect to the metrics $d_1$ and $d_2$ {{iff}}:
:for each neighborhood $N$ of $\map f a$ in $M_2$ there exists a corresponding neighborhood $N'$ of $a$ in $M_1$ such that $f \sqbrk {N'} \subseteq N$.
For a mapping $f: X \to Y$ we have:
:$f \sqbrk U \subseteq V \iff U \subset... | 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]] from $A_1$ to $A_2$.
Let $a \in A_1$ be a point in $A_1$.
$f$ is [[Definition:Continuous at Point of Metric Space|continuous at $a$]] with respect to t... | By definition, $f$ is [[Definition:Continuous at Point of Metric Space|continuous at $a$]] with respect to the [[Definition:Metric|metrics]] $d_1$ and $d_2$ {{iff}}:
:for each [[Definition:Neighborhood (Metric Space)|neighborhood]] $N$ of $\map f a$ in $M_2$ there exists a corresponding [[Definition:Neighborhood (Metri... | Metric Space Continuity by Inverse of Mapping between Neighborhoods | https://proofwiki.org/wiki/Metric_Space_Continuity_by_Inverse_of_Mapping_between_Neighborhoods | https://proofwiki.org/wiki/Metric_Space_Continuity_by_Inverse_of_Mapping_between_Neighborhoods | [
"Neighborhoods",
"Continuous Mappings on Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Mapping",
"Definition:Continuous Mapping (Metric Space)/Point",
"Definition:Metric Space/Metric",
"Definition:Neighborhood (Metric Space)",
"Definition:Neighborhood (Metric Space)"
] | [
"Definition:Continuous Mapping (Metric Space)/Point",
"Definition:Metric Space/Metric",
"Definition:Neighborhood (Metric Space)",
"Definition:Neighborhood (Metric Space)",
"Definition:Mapping"
] |
proofwiki-10187 | Point in Metric Space has Neighborhood | Let $M = \struct {A, d}$ be a metric space.
Let $a \in A$ be a point in $M$.
Then there exists some neighborhood of $a$ in $M$. | Let $a \in A$.
Then $A$ is a neighborhood of $a$ in $M$.
{{qed}} | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $a \in A$ be a point in $M$.
Then there exists some [[Definition:Neighborhood (Metric Space)|neighborhood]] of $a$ in $M$. | Let $a \in A$.
Then $A$ is a [[Definition:Neighborhood (Metric Space)|neighborhood]] of $a$ in $M$.
{{qed}} | Point in Metric Space has Neighborhood | https://proofwiki.org/wiki/Point_in_Metric_Space_has_Neighborhood | https://proofwiki.org/wiki/Point_in_Metric_Space_has_Neighborhood | [
"Neighborhoods"
] | [
"Definition:Metric Space",
"Definition:Neighborhood (Metric Space)"
] | [
"Definition:Neighborhood (Metric Space)"
] |
proofwiki-10188 | Point in Metric Space is Element of its Neighborhood | Let $N$ be a neighborhood of $a$ in $M$.
Then $a \in N$. | Trivially follows by definition of neighborhood of $a$.
{{qed}} | Let $N$ be a [[Definition:Neighborhood (Metric Space)|neighborhood]] of $a$ in $M$.
Then $a \in N$. | Trivially follows by definition of [[Definition:Neighborhood (Metric Space)|neighborhood]] of $a$.
{{qed}} | Point in Metric Space is Element of its Neighborhood | https://proofwiki.org/wiki/Point_in_Metric_Space_is_Element_of_its_Neighborhood | https://proofwiki.org/wiki/Point_in_Metric_Space_is_Element_of_its_Neighborhood | [
"Neighborhoods"
] | [
"Definition:Neighborhood (Metric Space)"
] | [
"Definition:Neighborhood (Metric Space)"
] |
proofwiki-10189 | Superset of Neighborhood in Metric Space is Neighborhood | Let $N$ be a neighborhood of $a$ in $M$.
Let $N \subseteq N' \subseteq A$.
Then $N'$ is a neighborhood of $a$ in $M$. | By definition of neighborhood:
:$\exists \epsilon \in \R_{>0}: \map {B_\epsilon} a \subseteq N$
where $\map {B_\epsilon} a$ is the open $\epsilon$-ball of $a$ in $M$.
By Subset Relation is Transitive:
:$\map {B_\epsilon} a \subseteq N'$
The result follows by definition of neighborhood of $a$.
{{qed}} | Let $N$ be a [[Definition:Neighborhood (Metric Space)|neighborhood]] of $a$ in $M$.
Let $N \subseteq N' \subseteq A$.
Then $N'$ is a [[Definition:Neighborhood (Metric Space)|neighborhood]] of $a$ in $M$. | By definition of [[Definition:Neighborhood (Metric Space)|neighborhood]]:
:$\exists \epsilon \in \R_{>0}: \map {B_\epsilon} a \subseteq N$
where $\map {B_\epsilon} a$ is the [[Definition:Open Ball|open $\epsilon$-ball]] of $a$ in $M$.
By [[Subset Relation is Transitive]]:
:$\map {B_\epsilon} a \subseteq N'$
The resu... | Superset of Neighborhood in Metric Space is Neighborhood | https://proofwiki.org/wiki/Superset_of_Neighborhood_in_Metric_Space_is_Neighborhood | https://proofwiki.org/wiki/Superset_of_Neighborhood_in_Metric_Space_is_Neighborhood | [
"Neighborhoods"
] | [
"Definition:Neighborhood (Metric Space)",
"Definition:Neighborhood (Metric Space)"
] | [
"Definition:Neighborhood (Metric Space)",
"Definition:Open Ball",
"Subset Relation is Transitive",
"Definition:Neighborhood (Metric Space)"
] |
proofwiki-10190 | Intersection of Neighborhoods in Metric Space is Neighborhood | Let $N, N'$ be neighborhoods of $a$ in $M$.
Then $N \cap N'$ is a neighborhood of $a$ in $M$. | By definition of neighborhood:
:$\exists \epsilon_1 \in \R_{>0}: \map {B_{\epsilon_1} } a \subseteq N$
where $\map {B_{\epsilon_1} } a$ is the open $\epsilon_1$-ball of $a$ in $M$.
:$\exists \epsilon_2 \in \R_{>0}: \map {B_{\epsilon_2} } a \subseteq N'$
where $\map {B_{\epsilon_2} } a$ is the open $\epsilon_2$-ball of ... | Let $N, N'$ be [[Definition:Neighborhood (Metric Space)|neighborhoods]] of $a$ in $M$.
Then $N \cap N'$ is a [[Definition:Neighborhood (Metric Space)|neighborhood]] of $a$ in $M$. | By definition of [[Definition:Neighborhood (Metric Space)|neighborhood]]:
:$\exists \epsilon_1 \in \R_{>0}: \map {B_{\epsilon_1} } a \subseteq N$
where $\map {B_{\epsilon_1} } a$ is the [[Definition:Open Ball|open $\epsilon_1$-ball]] of $a$ in $M$.
:$\exists \epsilon_2 \in \R_{>0}: \map {B_{\epsilon_2} } a \subseteq ... | Intersection of Neighborhoods in Metric Space is Neighborhood | https://proofwiki.org/wiki/Intersection_of_Neighborhoods_in_Metric_Space_is_Neighborhood | https://proofwiki.org/wiki/Intersection_of_Neighborhoods_in_Metric_Space_is_Neighborhood | [
"Neighborhoods"
] | [
"Definition:Neighborhood (Metric Space)",
"Definition:Neighborhood (Metric Space)"
] | [
"Definition:Neighborhood (Metric Space)",
"Definition:Open Ball",
"Definition:Open Ball",
"Definition:Set Intersection",
"Definition:Neighborhood (Metric Space)"
] |
proofwiki-10191 | Neighborhood in Metric Space has Subset Neighborhood | Let $N$ be a neighborhood of $a$ in $M$.
Then there exists a neighborhood $N'$ of $a$ such that:
:$(1): \quad N' \subseteq N$
:$(2): \quad N'$ is a neighborhood of each of its points. | By definition of neighborhood:
:$\exists \epsilon \in \R_{>0}: \map {B_\epsilon} a \subseteq N$
By Open Ball is Neighborhood of all Points Inside, $N' = \map {B_\epsilon} a$ fulfils the conditions of the statement.
{{qed}} | Let $N$ be a [[Definition:Neighborhood (Metric Space)|neighborhood]] of $a$ in $M$.
Then there exists a [[Definition:Neighborhood (Metric Space)|neighborhood]] $N'$ of $a$ such that:
:$(1): \quad N' \subseteq N$
:$(2): \quad N'$ is a [[Definition:Neighborhood (Metric Space)|neighborhood]] of each of its points. | By definition of [[Definition:Neighborhood (Metric Space)|neighborhood]]:
:$\exists \epsilon \in \R_{>0}: \map {B_\epsilon} a \subseteq N$
By [[Open Ball is Neighborhood of all Points Inside]], $N' = \map {B_\epsilon} a$ fulfils the conditions of the statement.
{{qed}} | Neighborhood in Metric Space has Subset Neighborhood | https://proofwiki.org/wiki/Neighborhood_in_Metric_Space_has_Subset_Neighborhood | https://proofwiki.org/wiki/Neighborhood_in_Metric_Space_has_Subset_Neighborhood | [
"Neighborhoods"
] | [
"Definition:Neighborhood (Metric Space)",
"Definition:Neighborhood (Metric Space)",
"Definition:Neighborhood (Metric Space)"
] | [
"Definition:Neighborhood (Metric Space)",
"Open Ball is Neighborhood of all Points Inside"
] |
proofwiki-10192 | Basis for Element of Real Number Line | Let $M = \struct {\R, d}$ denote the real number line with the usual (Euclidean) metric.
Let $a \in \R$ be a point in $M$.
Then the set of all open intervals containing $a$ is a basis for the neighborhood system of $a$. | 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 $a$.
The result follows from Open Ball in Real Number Line is Open Interval.
{{qed}} | Let $M = \struct {\R, d}$ 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 $M$.
Then the [[Definition:Set|set]] of all [[Definition:Open Real Interval|open intervals]] containing $a$ is a [[Definition:Basis for Neighborh... | 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 $a$]].
The result follows from [[Open Ball in Real Number Line is O... | Basis for Element of Real Number Line | https://proofwiki.org/wiki/Basis_for_Element_of_Real_Number_Line | https://proofwiki.org/wiki/Basis_for_Element_of_Real_Number_Line | [
"Real Intervals",
"Real Number Line with Euclidean Metric"
] | [
"Definition:Euclidean Metric/Real Number Line",
"Definition:Set",
"Definition:Real Interval/Open",
"Definition:Basis for Neighborhood System"
] | [
"Definition:Neighborhood (Metric Space)",
"Definition:Open Ball",
"Open Ball in Real Number Line is Open Interval"
] |
proofwiki-10193 | Open Ball in Real Number Line is Open Interval | Let $\struct {\R, d}$ denote the real number line $\R$ with the usual (Euclidean) metric $d$.
Let $x \in \R$ be a point in $\R$.
Let $\map {B_\epsilon} x$ be the open $\epsilon$-ball at $x$.
Then $\map {B_\epsilon} x$ is the open interval $\openint {x - \epsilon} {x + \epsilon}$. | Let $S = \map {B_\epsilon} x$ be an open $\epsilon$-ball at $x$.
Let $y \in \map {B_\epsilon} x$.
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 = {{Defof|Open Ball of Me... | Let $\struct {\R, d}$ denote the [[Definition:Real Number Line with Euclidean Metric|real number line $\R$ with the usual (Euclidean) metric $d$]].
Let $x \in \R$ be a point in $\R$.
Let $\map {B_\epsilon} x$ be the [[Definition:Open Ball of Metric Space|open $\epsilon$-ball at $x$]].
Then $\map {B_\epsilon} x$ is ... | Let $S = \map {B_\epsilon} x$ be an [[Definition:Open Ball of Metric Space|open $\epsilon$-ball at $x$]].
Let $y \in \map {B_\epsilon} x$.
Then:
{{begin-eqn}}
{{eqn | l = y
| o = \in
| m = \map {B_\epsilon} x
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \map d {y, x}
| o = <
| m = \... | Open Ball in Real Number Line is Open Interval | https://proofwiki.org/wiki/Open_Ball_in_Real_Number_Line_is_Open_Interval | https://proofwiki.org/wiki/Open_Ball_in_Real_Number_Line_is_Open_Interval | [
"Real Intervals",
"Open Balls",
"Real Number Line with Euclidean Metric"
] | [
"Definition:Euclidean Metric/Real Number Line",
"Definition:Open Ball",
"Definition:Real Interval/Open"
] | [
"Definition:Open Ball",
"Definition:Set Equality/Definition 2"
] |
proofwiki-10194 | Open Real Interval is Open Ball | Let $\R$ denote the real number line with the usual (Euclidean) metric.
Let $I := \openint a b \subseteq \R$ be an open real interval.
Then $I$ is the open $\epsilon$-ball $\map {B_\epsilon} \alpha$ of some $\alpha \in \R$. | Let:
{{begin-eqn}}
{{eqn | l = \alpha
| r = \frac {a + b} 2
}}
{{eqn | l = \epsilon
| r = \frac {b - a} 2
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = a
| r = \alpha - \epsilon
}}
{{eqn | l = b
| r = \alpha + \epsilon
}}
{{end-eqn}}
Thus:
:$\openint a b = \openint {\alpha - \epsilon} {\alpha +... | Let $\R$ denote the [[Definition:Real Number Line with Euclidean Metric|real number line with the usual (Euclidean) metric]].
Let $I := \openint a b \subseteq \R$ be an [[Definition:Open Real Interval|open real interval]].
Then $I$ is the [[Definition:Open Ball|open $\epsilon$-ball]] $\map {B_\epsilon} \alpha$ of so... | Let:
{{begin-eqn}}
{{eqn | l = \alpha
| r = \frac {a + b} 2
}}
{{eqn | l = \epsilon
| r = \frac {b - a} 2
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = a
| r = \alpha - \epsilon
}}
{{eqn | l = b
| r = \alpha + \epsilon
}}
{{end-eqn}}
Thus:
:$\openint a b = \openint {\alpha - \epsilon} {\a... | Open Real Interval is Open Ball | https://proofwiki.org/wiki/Open_Real_Interval_is_Open_Ball | https://proofwiki.org/wiki/Open_Real_Interval_is_Open_Ball | [
"Real Intervals",
"Open Balls",
"Real Number Line with Euclidean Metric"
] | [
"Definition:Euclidean Metric/Real Number Line",
"Definition:Real Interval/Open",
"Definition:Open Ball"
] | [
"Open Ball in Real Number Line is Open Interval",
"Definition:Open Ball"
] |
proofwiki-10195 | Neighborhoods in Standard Discrete Metric Space | Let $M = \struct {A, d}$ be a metric space where $d$ is the standard discrete metric.
Let $a \in A$.
Then $\set a$ is a neighborhood of $a$ which forms a basis for the system of neighborhoods of $a$. | By definition of the standard discrete metric:
:$\map d {x, y} = \begin {cases} 0 & : x = y \\ 1 & : x \ne y \end {cases}$
Let $\epsilon \in \R_{>0}$ such that $\epsilon < 1$.
Then:
{{begin-eqn}}
{{eqn | l = \map {B_\epsilon} a
| r = \set {x \in A: \map d {x, a} < \epsilon}
| c = {{Defof|Open Ball|Open $\e... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]] where $d$ is the [[Definition:Standard Discrete Metric|standard discrete metric]].
Let $a \in A$.
Then $\set a$ is a [[Definition:Neighborhood (Metric Space)|neighborhood]] of $a$ which forms a [[Definition:Basis for Neighborhood System|basis]] f... | By definition of the [[Definition:Standard Discrete Metric|standard discrete metric]]:
:$\map d {x, y} = \begin {cases} 0 & : x = y \\ 1 & : x \ne y \end {cases}$
Let $\epsilon \in \R_{>0}$ such that $\epsilon < 1$.
Then:
{{begin-eqn}}
{{eqn | l = \map {B_\epsilon} a
| r = \set {x \in A: \map d {x, a} < \eps... | Neighborhoods in Standard Discrete Metric Space | https://proofwiki.org/wiki/Neighborhoods_in_Standard_Discrete_Metric_Space | https://proofwiki.org/wiki/Neighborhoods_in_Standard_Discrete_Metric_Space | [
"Standard Discrete Metric",
"Neighborhoods"
] | [
"Definition:Metric Space",
"Definition:Standard Discrete Metric",
"Definition:Neighborhood (Metric Space)",
"Definition:Basis for Neighborhood System",
"Definition:System of Neighborhoods"
] | [
"Definition:Standard Discrete Metric",
"Definition:Neighborhood (Metric Space)",
"Definition:System of Neighborhoods",
"Definition:Neighborhood (Metric Space)",
"Point in Metric Space is Element of its Neighborhood",
"Definition:Basis for Neighborhood System"
] |
proofwiki-10196 | Subset of Standard Discrete Metric Space is Neighborhood of Each Point | Let $M = \struct {A, d}$ be a metric space where $d$ is the standard discrete metric.
Let $S \subseteq A$.
Let $a \in S$.
Then $S$ is a neighborhood of $a$.
That is, every subset of $A$ is a neighborhood of each of its points. | Let $S \subseteq A$.
Let $a \in S$.
From Neighborhoods in Standard Discrete Metric Space, $\set a$ is a neighborhood of $a$.
As $a \in S$ it follows from Singleton of Element is Subset that $\set a \subseteq S$.
The result follows from Superset of Neighborhood in Metric Space is Neighborhood.
{{qed}} | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]] where $d$ is the [[Definition:Standard Discrete Metric|standard discrete metric]].
Let $S \subseteq A$.
Let $a \in S$.
Then $S$ is a [[Definition:Neighborhood (Metric Space)|neighborhood]] of $a$.
That is, every [[Definition:Subset|subset]] of ... | Let $S \subseteq A$.
Let $a \in S$.
From [[Neighborhoods in Standard Discrete Metric Space]], $\set a$ is a [[Definition:Neighborhood (Metric Space)|neighborhood]] of $a$.
As $a \in S$ it follows from [[Singleton of Element is Subset]] that $\set a \subseteq S$.
The result follows from [[Superset of Neighborhood in... | Subset of Standard Discrete Metric Space is Neighborhood of Each Point | https://proofwiki.org/wiki/Subset_of_Standard_Discrete_Metric_Space_is_Neighborhood_of_Each_Point | https://proofwiki.org/wiki/Subset_of_Standard_Discrete_Metric_Space_is_Neighborhood_of_Each_Point | [
"Standard Discrete Metric",
"Neighborhoods"
] | [
"Definition:Metric Space",
"Definition:Standard Discrete Metric",
"Definition:Neighborhood (Metric Space)",
"Definition:Subset",
"Definition:Neighborhood (Metric Space)"
] | [
"Neighborhoods in Standard Discrete Metric Space",
"Definition:Neighborhood (Metric Space)",
"Singleton of Element is Subset",
"Superset of Neighborhood in Metric Space is Neighborhood"
] |
proofwiki-10197 | Continuity of Heaviside Step Function | Let $\mu_c: \R \to \R$ be the Heaviside step function:
:$\map {\mu_c} x = \begin {cases} 0 & : x < c \\ 1 & : x > c \\ \text {arbitrary} & : x = c \end {cases}$
Then $\mu_c$ is continuous at every point of $\R$ except at $c$. | Let $x \in \R: x \ne c$.
Let $\epsilon \in \R_{>0}$.
Let $\delta < \size {x - c}$.
Then by definition of the Heaviside step function:
:$\forall y \in \closedint {x - \delta} {x + \delta}: \map {\mu_c} x = \begin {cases} 0 & : x < c \\ 1 & : x > c \end {cases}$
Thus:
:$\forall y \in \closedint {x - \delta} {x + \delta}:... | Let $\mu_c: \R \to \R$ be the [[Definition:Heaviside Step Function|Heaviside step function]]:
:$\map {\mu_c} x = \begin {cases} 0 & : x < c \\ 1 & : x > c \\ \text {arbitrary} & : x = c \end {cases}$
Then $\mu_c$ is [[Definition:Continuous Real Function at Point|continuous]] at every point of $\R$ except at $c$. | Let $x \in \R: x \ne c$.
Let $\epsilon \in \R_{>0}$.
Let $\delta < \size {x - c}$.
Then by definition of the [[Definition:Heaviside Step Function|Heaviside step function]]:
:$\forall y \in \closedint {x - \delta} {x + \delta}: \map {\mu_c} x = \begin {cases} 0 & : x < c \\ 1 & : x > c \end {cases}$
Thus:
:$\forall ... | Continuity of Heaviside Step Function | https://proofwiki.org/wiki/Continuity_of_Heaviside_Step_Function | https://proofwiki.org/wiki/Continuity_of_Heaviside_Step_Function | [
"Continuity",
"Heaviside Step Function"
] | [
"Definition:Heaviside Step Function",
"Definition:Continuous Real Function/Point"
] | [
"Definition:Heaviside Step Function",
"Definition:Continuous Real Function/Point",
"Definition:Continuous Real Function/Point"
] |
proofwiki-10198 | Metric Space Continuity by Neighborhood Basis | 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 from $A_1$ to $A_2$.
Let $a \in A_1$ be a point in $A_1$.
Let $\BB_{\map f a}$ be a basis for the neighborhood system at $\map f a$.
$f$ is continuous at $a$ with respect to the metrics $d_1$ and $d_2$ {{if... | By definition, $\BB_{\map f a}$ be a basis for the neighborhood system at $\map f a$ {{iff}}:
:$\forall N_a \subseteq M_2: \exists N \in \BB_{\map f a}: N \subseteq N_a$
where $N_a$ denotes a neighborhood of $\map f a$ in $M_2$. | 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]] from $A_1$ to $A_2$.
Let $a \in A_1$ be a point in $A_1$.
Let $\BB_{\map f a}$ be a [[Definition:Basis for Neighborhood System|basis for the neighborhoo... | By definition, $\BB_{\map f a}$ be a [[Definition:Basis for Neighborhood System|basis for the neighborhood system at $\map f a$]] {{iff}}:
:$\forall N_a \subseteq M_2: \exists N \in \BB_{\map f a}: N \subseteq N_a$
where $N_a$ denotes a [[Definition:Neighborhood (Metric Space)|neighborhood]] of $\map f a$ in $M_2$. | Metric Space Continuity by Neighborhood Basis | https://proofwiki.org/wiki/Metric_Space_Continuity_by_Neighborhood_Basis | https://proofwiki.org/wiki/Metric_Space_Continuity_by_Neighborhood_Basis | [
"Neighborhood Bases",
"Continuous Mappings on Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Mapping",
"Definition:Basis for Neighborhood System",
"Definition:Continuous Mapping (Metric Space)/Point",
"Definition:Metric Space/Metric",
"Definition:Neighborhood (Metric Space)",
"Definition:Neighborhood (Metric Space)"
] | [
"Definition:Basis for Neighborhood System",
"Definition:Neighborhood (Metric Space)",
"Definition:Neighborhood (Metric Space)",
"Definition:Neighborhood (Metric Space)",
"Definition:Neighborhood (Metric Space)",
"Definition:Neighborhood (Metric Space)",
"Definition:Neighborhood (Metric Space)",
"Defin... |
proofwiki-10199 | Closed Intervals 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:
:$\BB_a := \set {\closedint {a - \epsilon} {a + \epsilon}: \epsilon \in \R_{>0} }$
that is, the set of all closed intervals of $\R$ with $a$ as a midpoint.
Then $\BB_a$ is a basis for the ne... | 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:
:$\BB_a := \set {\closedint {a - \epsilon} {a + \epsilon}: \epsilon \in \R_{>0} }$
that is, the [[Definition:Set|set]] of all [[Defin... | 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... | Closed Intervals form Neighborhood Basis in Real Number Line | https://proofwiki.org/wiki/Closed_Intervals_form_Neighborhood_Basis_in_Real_Number_Line | https://proofwiki.org/wiki/Closed_Intervals_form_Neighborhood_Basis_in_Real_Number_Line | [
"Real Intervals",
"Real Number Line with Euclidean Metric"
] | [
"Definition:Euclidean Metric/Real Number Line",
"Definition:Set",
"Definition:Real Interval/Closed",
"Definition:Real Interval/Midpoint",
"Definition:Basis for Neighborhood System"
] | [
"Definition:Neighborhood (Metric Space)",
"Definition:Open Ball",
"Open Ball in Real Number Line is Open Interval",
"Definition:Real Interval/Closed",
"Definition:Real Interval/Open",
"Subset Relation is Transitive",
"Definition:Real Interval/Closed",
"Definition:Real Interval/Open",
"Open Real Inte... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.