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-11500 | Intersection of Real Intervals is Real Interval | Let $I_1$ and $I_2$ be real intervals.
Then $I_1 \cap I_2$ is also a real interval. | Let $x, y \in I_1 \cap I_2$.
From the definition of a real interval, it suffices to show that:
:for each $z \in \R$ with $x \le z \le y$ we have $z \in I_1 \cap I_2$.
Let $z$ be a real number with:
:$x \le z \le y$
Since $x, y \in I_1$, we have:
:$z \in I_1$
from the definition of a real interval.
Similarly, since $x... | Let $I_1$ and $I_2$ be [[Definition:Real Interval|real intervals]].
Then $I_1 \cap I_2$ is also a [[Definition:Real Interval|real interval]]. | Let $x, y \in I_1 \cap I_2$.
From the definition of a [[Definition:Real Interval/Definition 1|real interval]], it suffices to show that:
:for each $z \in \R$ with $x \le z \le y$ we have $z \in I_1 \cap I_2$.
Let $z$ be a [[Definition:Real Number|real number]] with:
:$x \le z \le y$
Since $x, y \in I_1$, we have:... | Intersection of Real Intervals is Real Interval | https://proofwiki.org/wiki/Intersection_of_Real_Intervals_is_Real_Interval | https://proofwiki.org/wiki/Intersection_of_Real_Intervals_is_Real_Interval | [
"Real Intervals"
] | [
"Definition:Real Interval",
"Definition:Real Interval"
] | [
"Definition:Real Interval/Definition 1",
"Definition:Real Number",
"Definition:Real Interval/Definition 1",
"Definition:Real Interval"
] |
proofwiki-11501 | Union of Real Intervals is not necessarily Real Interval | Let $I_1$ and $I_2$ be real intervals.
Then $I_1 \cup I_2$ is not necessarily a real interval. | Proof by Counterexample:
Consider the real intervals:
:$I_1 = \left({0 \,.\,.\, 2}\right)$
:$I_2 = \left({4 \,.\,.\, 6}\right)$
Then we have that:
:$1 < 3 < 5$
where:
:$1 \in I_1 \cup I_2$
:$5 \in I_1 \cup I_2$
but:
:$3 \notin I_1 \cup I_2$
Thus $I_1 \cup I_2$ is not a real interval.
{{qed}} | Let $I_1$ and $I_2$ be [[Definition:Real Interval|real intervals]].
Then $I_1 \cup I_2$ is not necessarily a [[Definition:Real Interval|real interval]]. | [[Proof by Counterexample]]:
Consider the [[Definition:Real Interval|real intervals]]:
:$I_1 = \left({0 \,.\,.\, 2}\right)$
:$I_2 = \left({4 \,.\,.\, 6}\right)$
Then we have that:
:$1 < 3 < 5$
where:
:$1 \in I_1 \cup I_2$
:$5 \in I_1 \cup I_2$
but:
:$3 \notin I_1 \cup I_2$
Thus $I_1 \cup I_2$ is not a [[Definition:R... | Union of Real Intervals is not necessarily Real Interval | https://proofwiki.org/wiki/Union_of_Real_Intervals_is_not_necessarily_Real_Interval | https://proofwiki.org/wiki/Union_of_Real_Intervals_is_not_necessarily_Real_Interval | [
"Real Intervals"
] | [
"Definition:Real Interval",
"Definition:Real Interval"
] | [
"Proof by Counterexample",
"Definition:Real Interval",
"Definition:Real Interval"
] |
proofwiki-11502 | Domain of Real Square Function | The domain of the real square function is the entire set of real numbers $\R$. | The operation of real multiplication is defined on all real numbers.
Thus:
:$\forall x \in \R: \exists y \in \R: x^2 = y$
Hence the result by definition of domain.
{{qed}} | The [[Definition:Domain of Real Function|domain]] of the [[Definition:Real Square Function|real square function]] is the entire [[Definition:Real Number|set of real numbers]] $\R$. | The operation of [[Definition:Real Multiplication|real multiplication]] is defined on all [[Definition:Real Number|real numbers]].
Thus:
:$\forall x \in \R: \exists y \in \R: x^2 = y$
Hence the result by definition of [[Definition:Domain of Real Function|domain]].
{{qed}} | Domain of Real Square Function | https://proofwiki.org/wiki/Domain_of_Real_Square_Function | https://proofwiki.org/wiki/Domain_of_Real_Square_Function | [
"Square Function"
] | [
"Definition:Real Function/Domain",
"Definition:Square Function/Real",
"Definition:Real Number"
] | [
"Definition:Multiplication/Real Numbers",
"Definition:Real Number",
"Definition:Real Function/Domain"
] |
proofwiki-11503 | Image of Real Square Function | The image of the real square function is the entire set of positive real numbers $\R_{\ge 0}$. | From Square of Real Number is Non-Negative, the image of $f$ is $\R_{\ge 0}$.
From Positive Real has Real Square Root:
:$\forall x \in \R: \exists y \in \R: x^2 = y$
Hence the result by definition of image.
{{qed}} | The [[Definition:Image of Mapping|image]] of the [[Definition:Real Square Function|real square function]] is the entire [[Definition:Positive Real Number|set of positive real numbers]] $\R_{\ge 0}$. | From [[Square of Real Number is Non-Negative]], the [[Definition:Image of Mapping|image]] of $f$ is $\R_{\ge 0}$.
From [[Positive Real has Real Square Root]]:
:$\forall x \in \R: \exists y \in \R: x^2 = y$
Hence the result by definition of [[Definition:Image of Mapping|image]].
{{qed}} | Image of Real Square Function | https://proofwiki.org/wiki/Image_of_Real_Square_Function | https://proofwiki.org/wiki/Image_of_Real_Square_Function | [
"Square Function"
] | [
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Square Function/Real",
"Definition:Positive/Real Number"
] | [
"Square of Real Number is Non-Negative",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Positive Real has Real Square Root",
"Definition:Image (Set Theory)/Mapping/Mapping"
] |
proofwiki-11504 | Domain of Real Square Root Function | The domain of the real square root function is the set of positive real numbers $\R_{\ge 0}$:
:$\set{x \in \R: x \ge 0}$ | From Square of Real Number is Non-Negative:
:$\forall x \in \R: x^2 \ge 0$
Hence the result by definition of domain.
{{qed}} | The [[Definition:Domain of Real Function|domain]] of the [[Definition:Real Square Root Function|real square root function]] is the [[Definition:Positive Real Number|set of positive real numbers]] $\R_{\ge 0}$:
:$\set{x \in \R: x \ge 0}$ | From [[Square of Real Number is Non-Negative]]:
:$\forall x \in \R: x^2 \ge 0$
Hence the result by definition of [[Definition:Domain of Real Function|domain]].
{{qed}} | Domain of Real Square Root Function | https://proofwiki.org/wiki/Domain_of_Real_Square_Root_Function | https://proofwiki.org/wiki/Domain_of_Real_Square_Root_Function | [
"Real Functions"
] | [
"Definition:Real Function/Domain",
"Real Function/Examples/Square Root",
"Definition:Positive/Real Number"
] | [
"Square of Real Number is Non-Negative",
"Definition:Real Function/Domain"
] |
proofwiki-11505 | Exponent Combination Laws/Product of Powers/Proof 1 | Let $a \in \R_{> 0}$ be a positive real number.
Let $x, y \in \R$ be real numbers.
Let $a^x$ be defined as $a$ to the power of $x$.
Then:
:$a^x a^y = a^{x + y}$ | {{begin-eqn}}
{{eqn | l = a^{x + y}
| r = \map \exp {\paren {x + y} \ln a}
| c = {{Defof|Power to Real Number}}
}}
{{eqn | r = \map \exp {x \ln a + y \ln a}
| c =
}}
{{eqn | r = \map \exp {x \ln a} \, \map \exp {y \ln a}
| c = Exponential of Sum
}}
{{eqn | r = a^x a^y
| c = {{Defof|Power ... | Let $a \in \R_{> 0}$ be a [[Definition:Positive Real Number|positive real number]].
Let $x, y \in \R$ be [[Definition:Real Number|real numbers]].
Let $a^x$ be defined as [[Definition:Power to Real Number|$a$ to the power of $x$]].
Then:
:$a^x a^y = a^{x + y}$ | {{begin-eqn}}
{{eqn | l = a^{x + y}
| r = \map \exp {\paren {x + y} \ln a}
| c = {{Defof|Power to Real Number}}
}}
{{eqn | r = \map \exp {x \ln a + y \ln a}
| c =
}}
{{eqn | r = \map \exp {x \ln a} \, \map \exp {y \ln a}
| c = [[Exponential of Sum]]
}}
{{eqn | r = a^x a^y
| c = {{Defof|Po... | Exponent Combination Laws/Product of Powers/Proof 1 | https://proofwiki.org/wiki/Exponent_Combination_Laws/Product_of_Powers/Proof_1 | https://proofwiki.org/wiki/Exponent_Combination_Laws/Product_of_Powers/Proof_1 | [
"Exponent Combination Laws"
] | [
"Definition:Positive/Real Number",
"Definition:Real Number",
"Definition:Power (Algebra)/Real Number"
] | [
"Exponential of Sum"
] |
proofwiki-11506 | 2197 is Cube of 13 | :$13^3 = 2197$ | By long multiplication:
<pre>
13
x 13
-----
39
130
-----
169
-----
</pre>
then:
<pre>
x 169
13
------
507
1690
-----
2197
------
</pre>
{{qed}} | :$13^3 = 2197$ | By [[Definition:Long Multiplication|long multiplication]]:
<pre>
13
x 13
-----
39
130
-----
169
-----
</pre>
then:
<pre>
x 169
13
------
507
1690
-----
2197
------
</pre>
{{qed}} | 2197 is Cube of 13 | https://proofwiki.org/wiki/2197_is_Cube_of_13 | https://proofwiki.org/wiki/2197_is_Cube_of_13 | [
"Numbers"
] | [] | [
"Definition:Long Multiplication"
] |
proofwiki-11507 | Inequality of Product of Unequal Numbers | Let $a, b, c, d \in \R$.
Then:
:$0 < a < b \land 0 < c < d \implies 0 < a c < b d$ | {{begin-eqn}}
{{eqn | n = 1
| o =
| r = 0 < a < b
| c =
}}
{{eqn | n = 2
| o = \leadsto
| r = 0 < b
| c = Ordering is Transitive
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | n = 3
| o =
| r = 0 < c < d
| c =
}}
{{eqn | n = 4
| o = \leadsto
| r = 0 < c
... | Let $a, b, c, d \in \R$.
Then:
:$0 < a < b \land 0 < c < d \implies 0 < a c < b d$ | {{begin-eqn}}
{{eqn | n = 1
| o =
| r = 0 < a < b
| c =
}}
{{eqn | n = 2
| o = \leadsto
| r = 0 < b
| c = [[Definition:Ordering/Definition 1|Ordering]] is [[Definition:Transitive|Transitive]]
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | n = 3
| o =
| r = 0 < c < d
| c ... | Inequality of Product of Unequal Numbers | https://proofwiki.org/wiki/Inequality_of_Product_of_Unequal_Numbers | https://proofwiki.org/wiki/Inequality_of_Product_of_Unequal_Numbers | [
"Inequalities",
"Real Multiplication"
] | [] | [
"Definition:Ordering/Definition 1",
"Definition:Transitive",
"Definition:Ordering/Definition 1",
"Definition:Transitive",
"Definition:Ordering/Definition 1",
"Definition:Transitive",
"Category:Inequalities",
"Category:Real Multiplication"
] |
proofwiki-11508 | Negative of Absolute Value/Corollary 3 | Let $y \in \R_{\ge 0}$.
Let $z \in \R$.
Then:
:$\size {x - z} < y \iff z - y < x < z + y$ | {{begin-eqn}}
{{eqn | l = \size {x - z}
| o = <
| m = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = -y
| o = <
| m = x - z
| mo= <
| r = y
| c = {{Corollary|Negative of Absolute Value|1}}
}}
{{eqn | ll= \leadstoandfrom
| l = z - y
| o = <
| m = x
... | Let $y \in \R_{\ge 0}$.
Let $z \in \R$.
Then:
:$\size {x - z} < y \iff z - y < x < z + y$ | {{begin-eqn}}
{{eqn | l = \size {x - z}
| o = <
| m = y
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = -y
| o = <
| m = x - z
| mo= <
| r = y
| c = {{Corollary|Negative of Absolute Value|1}}
}}
{{eqn | ll= \leadstoandfrom
| l = z - y
| o = <
| m = x
... | Negative of Absolute Value/Corollary 3 | https://proofwiki.org/wiki/Negative_of_Absolute_Value/Corollary_3 | https://proofwiki.org/wiki/Negative_of_Absolute_Value/Corollary_3 | [
"Negative of Absolute Value"
] | [] | [
"Real Number Ordering is Compatible with Addition",
"Category:Negative of Absolute Value"
] |
proofwiki-11509 | Exponent Combination Laws/Product of Powers/Proof 2/Lemma | Let $x_1, x_2, y_1, y_2 \in \R_{>0}$ be strictly positive real numbers.
Let $\epsilon \in \openint 0 {\min \set {y_1, y_2, 1} }$.
Then:
:$\size {x_1 - y_1} < \epsilon \land \size {x_2 - y_2} < \epsilon \implies \size {x_1 x_2 - y_1 y_2} < \epsilon \paren {y_1 + y_2 + 1}$ | First:
{{begin-eqn}}
{{eqn | l = \epsilon
| o = <
| r = \min \set {y_1, y_2, 1}
}}
{{eqn | ll= \leadsto
| l = \epsilon
| o = <
| r = y_1
| c = {{Defof|Min Operation}}
}}
{{eqn | ll= \leadsto
| l = \epsilon - \epsilon
| o = <
| r = y_1 - \epsilon
| c = subtract... | Let $x_1, x_2, y_1, y_2 \in \R_{>0}$ be [[Definition:Strictly Positive Real Number|strictly positive real numbers]].
Let $\epsilon \in \openint 0 {\min \set {y_1, y_2, 1} }$.
Then:
:$\size {x_1 - y_1} < \epsilon \land \size {x_2 - y_2} < \epsilon \implies \size {x_1 x_2 - y_1 y_2} < \epsilon \paren {y_1 + y_2 + 1}$ | First:
{{begin-eqn}}
{{eqn | l = \epsilon
| o = <
| r = \min \set {y_1, y_2, 1}
}}
{{eqn | ll= \leadsto
| l = \epsilon
| o = <
| r = y_1
| c = {{Defof|Min Operation}}
}}
{{eqn | ll= \leadsto
| l = \epsilon - \epsilon
| o = <
| r = y_1 - \epsilon
| c = subtract... | Exponent Combination Laws/Product of Powers/Proof 2/Lemma | https://proofwiki.org/wiki/Exponent_Combination_Laws/Product_of_Powers/Proof_2/Lemma | https://proofwiki.org/wiki/Exponent_Combination_Laws/Product_of_Powers/Proof_2/Lemma | [
"Exponent Combination Laws"
] | [
"Definition:Strictly Positive/Real Number"
] | [
"Inequality of Product of Unequal Numbers",
"Distributive Laws/Arithmetic",
"Square of Non-Zero Real Number is Strictly Positive",
"Real Number between Zero and One is Greater than Square",
"Distributive Laws/Arithmetic",
"Category:Exponent Combination Laws"
] |
proofwiki-11510 | Union of Unordered Tuples | Let $x_1, \dots, x_n, x_{n+1}, \dots, x_m$ be arbitrary.
Then
:$\set {x_1, \dots, x_n} \cup \set {x_{n + 1}, \dots, x_m} = \set {x_1, \dots, x_n, x_{n + 1}, \dots, x_m}$ | Let $a$ be arbitrary.
:$a \in \set {x_1, \dots, x_n} \cup \set {x_{n + 1}, \dots, x_m}$
{{iff}}
:$a \in \set {x_1, \dots, x_n}$ or $a \in \set {x_{n + 1}, \dots, x_m}$ by definition of union
{{iff}}
:$a = x_1 \lor \dots \lor a = x_n$ or $a = x_{n + 1} \lor \dots \lor a = x_m$ by definition of unordered tuple
{{iff}}
:$... | Let $x_1, \dots, x_n, x_{n+1}, \dots, x_m$ be arbitrary.
Then
:$\set {x_1, \dots, x_n} \cup \set {x_{n + 1}, \dots, x_m} = \set {x_1, \dots, x_n, x_{n + 1}, \dots, x_m}$ | Let $a$ be arbitrary.
:$a \in \set {x_1, \dots, x_n} \cup \set {x_{n + 1}, \dots, x_m}$
{{iff}}
:$a \in \set {x_1, \dots, x_n}$ or $a \in \set {x_{n + 1}, \dots, x_m}$ by definition of [[Definition:Set Union|union]]
{{iff}}
:$a = x_1 \lor \dots \lor a = x_n$ or $a = x_{n + 1} \lor \dots \lor a = x_m$ by definition of ... | Union of Unordered Tuples | https://proofwiki.org/wiki/Union_of_Unordered_Tuples | https://proofwiki.org/wiki/Union_of_Unordered_Tuples | [
"Set Theory"
] | [] | [
"Definition:Set Union",
"Definition:Unordered Tuple",
"Definition:Unordered Tuple",
"Definition:Set Equality",
"Category:Set Theory"
] |
proofwiki-11511 | Gauss-Lucas Theorem | Let $P$ be a non-constant polynomial in $\C$.
Then all zeroes of its derivative $P'$ belong to the convex hull of the set of zeroes of $P$. | Over the complex numbers, $P$ is a product of prime factors:
:$\ds \map P z = \alpha \prod_{i \mathop = 1}^n \paren{ z - a_i }$
where:
:$a_1, a_2, \ldots, a_n \in \C$ are the (not necessary distinct) zeroes of $P$
:$\alpha \in \C$ is the leading coefficient of $P$
:$n$ is the degree of $P$.
Let $z$ be any complex numbe... | Let $P$ be a non-[[Definition:Constant Polynomial|constant]] [[Definition:Polynomial over Complex Numbers|polynomial]] in $\C$.
Then all [[Definition:Zero of Function|zeroes]] of its [[Definition:Derivative of Complex Function|derivative]] $P'$ belong to the [[Definition:Convex Hull|convex hull]] of the set of [[Defin... | Over the [[Definition:Complex Numbers|complex numbers]], $P$ is a product of prime factors:
:$\ds \map P z = \alpha \prod_{i \mathop = 1}^n \paren{ z - a_i }$
where:
:$a_1, a_2, \ldots, a_n \in \C$ are the (not necessary distinct) [[Definition:Zero of Function|zeroes]] of $P$
:$\alpha \in \C$ is the [[Definition:Lead... | Gauss-Lucas Theorem | https://proofwiki.org/wiki/Gauss-Lucas_Theorem | https://proofwiki.org/wiki/Gauss-Lucas_Theorem | [
"Complex Analysis"
] | [
"Definition:Constant Polynomial",
"Definition:Polynomial/Complex Numbers",
"Definition:Root of Mapping",
"Definition:Derivative/Complex Function",
"Definition:Convex Hull",
"Definition:Root of Mapping"
] | [
"Definition:Complex Number",
"Definition:Root of Mapping",
"Definition:Leading Coefficient",
"Definition:Degree of Polynomial",
"Definition:Complex Number",
"Definition:Logarithmic Derivative",
"Definition:Root of Mapping",
"Definition:Barycentric Coordinates (Astronomy)",
"Definition:Convex Combina... |
proofwiki-11512 | Way Below implies Preceding | Let $\left({S, \preceq}\right)$ be an ordered set.
Let $x, y \in S$ such that
:$x \ll y$
where $\ll$ denotes element is way below second element.
Then
:$x \preceq y$ | By Singleton is Directed and Filtered Subset:
:$\left\{ {y}\right\}$ is directed.
By Supremum of Singleton:
:$\left\{ {y}\right\}$ admits a supremum and $\sup \left\{ {y}\right\} = y$
By definition of reflexivity:
:$y \preceq \sup \left\{ {y}\right\}$
By definition of way below:
:$\exists d \in \left\{ {y}\right\}: x \... | Let $\left({S, \preceq}\right)$ be an [[Definition:Ordered Set|ordered set]].
Let $x, y \in S$ such that
:$x \ll y$
where $\ll$ denotes [[Definition:Element is Way Below|element is way below second element]].
Then
:$x \preceq y$ | By [[Singleton is Directed and Filtered Subset]]:
:$\left\{ {y}\right\}$ is [[Definition:Directed Subset|directed]].
By [[Supremum of Singleton]]:
:$\left\{ {y}\right\}$ admits a [[Definition:Supremum of Set|supremum]] and $\sup \left\{ {y}\right\} = y$
By definition of [[Definition:Reflexivity|reflexivity]]:
:$y \pr... | Way Below implies Preceding | https://proofwiki.org/wiki/Way_Below_implies_Preceding | https://proofwiki.org/wiki/Way_Below_implies_Preceding | [
"Way Below Relation"
] | [
"Definition:Ordered Set",
"Definition:Element is Way Below"
] | [
"Singleton is Directed and Filtered Subset",
"Definition:Directed Subset",
"Supremum of Singleton",
"Definition:Supremum of Set",
"Definition:Reflexivity",
"Definition:Element is Way Below",
"Definition:Singleton"
] |
proofwiki-11513 | Preceding and Way Below implies Way Below | Let $\struct {S, \preceq}$ be an ordered set.
Let $u, x, y, z \in S$ such that
:$u \preceq x \ll y \preceq z$
where $\ll$ denotes the way below relation.
Then
:$u \ll z$ | Let $D$ be a directed subset of $S$ such that
:$D$ admits a supremum
and
:$z \preceq \sup D$
By definition of transitivity:
:$y \preceq \sup D$
By definition of way below relation:
:$\exists d \in D: x \preceq d$
Thus by definition of transitivity:
:$\exists d \in D: u \preceq d$
Thus by definition of way below relatio... | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $u, x, y, z \in S$ such that
:$u \preceq x \ll y \preceq z$
where $\ll$ denotes the [[Definition:Element is Way Below|way below relation]].
Then
:$u \ll z$ | Let $D$ be a [[Definition:Directed Subset|directed subset]] of $S$ such that
:$D$ admits a [[Definition:Supremum of Set|supremum]]
and
:$z \preceq \sup D$
By definition of [[Definition:Transitivity|transitivity]]:
:$y \preceq \sup D$
By definition of [[Definition:Element is Way Below|way below relation]]:
:$\exists d... | Preceding and Way Below implies Way Below | https://proofwiki.org/wiki/Preceding_and_Way_Below_implies_Way_Below | https://proofwiki.org/wiki/Preceding_and_Way_Below_implies_Way_Below | [
"Way Below Relation"
] | [
"Definition:Ordered Set",
"Definition:Element is Way Below"
] | [
"Definition:Directed Subset",
"Definition:Supremum of Set",
"Definition:Transitive",
"Definition:Element is Way Below",
"Definition:Transitive",
"Definition:Element is Way Below"
] |
proofwiki-11514 | Join is Way Below if Operands are Way Below | Let $\struct {S, \vee, \preceq}$ be a join semilattice.
Let $x, y, z \in S$ such that
:$x \ll z$ and $y \ll z$
where $\ll$ denotes the way below relation.
Then
:$x \vee y \ll z$ | Let $D$ be a directed subset of $S$ such that
:$D$ admits a supremum
and
:$z \preceq \sup D$
By definition of way below relation:
:$\exists d_1 \in D: x \preceq d_1$
and
:$\exists d_2 \in D: y \preceq d_2$
By definition of directed subset:
:$\exists d \in D: d_1 \preceq d$ and $d_2 \preceq d$
By definition of transitiv... | Let $\struct {S, \vee, \preceq}$ be a [[Definition:Join Semilattice|join semilattice]].
Let $x, y, z \in S$ such that
:$x \ll z$ and $y \ll z$
where $\ll$ denotes the [[Definition:Element is Way Below|way below relation]].
Then
:$x \vee y \ll z$ | Let $D$ be a [[Definition:Directed Subset|directed subset]] of $S$ such that
:$D$ admits a [[Definition:Supremum of Set|supremum]]
and
:$z \preceq \sup D$
By definition of [[Definition:Element is Way Below|way below relation]]:
:$\exists d_1 \in D: x \preceq d_1$
and
:$\exists d_2 \in D: y \preceq d_2$
By definition ... | Join is Way Below if Operands are Way Below | https://proofwiki.org/wiki/Join_is_Way_Below_if_Operands_are_Way_Below | https://proofwiki.org/wiki/Join_is_Way_Below_if_Operands_are_Way_Below | [
"Way Below Relation"
] | [
"Definition:Join Semilattice",
"Definition:Element is Way Below"
] | [
"Definition:Directed Subset",
"Definition:Supremum of Set",
"Definition:Element is Way Below",
"Definition:Directed Subset",
"Definition:Transitive",
"Definition:Supremum of Set",
"Definition:Element is Way Below"
] |
proofwiki-11515 | Way Below Relation is Transitive | Let $\left({S, \preceq}\right)$ be an ordered set.
Let $x, y, z \in S$ such that
:$x \ll y \ll z$
Then
:$x \ll z$ | By Way Below implies Preceding:
:$x \preceq y$
By definition of reflexivity:
:$z \preceq z$
Thus by Preceding and Way Below implies Way Below:
:$x \ll z$
{{qed}} | Let $\left({S, \preceq}\right)$ be an [[Definition:Ordered Set|ordered set]].
Let $x, y, z \in S$ such that
:$x \ll y \ll z$
Then
:$x \ll z$ | By [[Way Below implies Preceding]]:
:$x \preceq y$
By definition of [[Definition:Reflexivity|reflexivity]]:
:$z \preceq z$
Thus by [[Preceding and Way Below implies Way Below]]:
:$x \ll z$
{{qed}} | Way Below Relation is Transitive | https://proofwiki.org/wiki/Way_Below_Relation_is_Transitive | https://proofwiki.org/wiki/Way_Below_Relation_is_Transitive | [
"Way Below Relation"
] | [
"Definition:Ordered Set"
] | [
"Way Below implies Preceding",
"Definition:Reflexivity",
"Preceding and Way Below implies Way Below"
] |
proofwiki-11516 | Way Below Relation is Antisymmetric | Let $\struct {S, \preceq}$ be an ordered set.
Let $x, y \in S$ such that
:$x \ll y$ and $y \ll x$
Then
:$x = y$ | By Way Below implies Preceding:
:$x \preceq y$ and $y \preceq x$
Thus by definition of antisymmetry:
:$x = y$
{{qed}} | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $x, y \in S$ such that
:$x \ll y$ and $y \ll x$
Then
:$x = y$ | By [[Way Below implies Preceding]]:
:$x \preceq y$ and $y \preceq x$
Thus by definition of [[Definition:Antisymmetric Relation|antisymmetry]]:
:$x = y$
{{qed}} | Way Below Relation is Antisymmetric | https://proofwiki.org/wiki/Way_Below_Relation_is_Antisymmetric | https://proofwiki.org/wiki/Way_Below_Relation_is_Antisymmetric | [
"Way Below Relation"
] | [
"Definition:Ordered Set"
] | [
"Way Below implies Preceding",
"Definition:Antisymmetric Relation"
] |
proofwiki-11517 | Continued Fraction Expansion of Golden Mean | The golden mean has the simplest possible continued fraction expansion, namely $\sqbrk {1, 1, 1, 1, \ldots}$:
:$\phi = 1 + \cfrac 1 {1 + \cfrac 1 {1 + \cfrac 1 {\ddots} } }$ | Let:
:$x = 1 + \cfrac 1 {1 + \cfrac 1 {1 + \cfrac 1 {\ddots} } }$
Then:
{{begin-eqn}}
{{eqn | l = x
| r = 1 + \frac 1 x
| c = substituting for $x$
}}
{{eqn | ll= \leadsto
| l = x^2
| r = x + 1
| c =
}}
{{eqn | ll= \leadsto
| l = x^2 - x - 1
| r = 0
| c =
}}
{{end-eqn}}
... | The [[Definition:Golden Mean|golden mean]] has the simplest possible [[Definition:Continued Fraction Expansion of Irrational Number|continued fraction expansion]], namely $\sqbrk {1, 1, 1, 1, \ldots}$:
:$\phi = 1 + \cfrac 1 {1 + \cfrac 1 {1 + \cfrac 1 {\ddots} } }$ | Let:
:$x = 1 + \cfrac 1 {1 + \cfrac 1 {1 + \cfrac 1 {\ddots} } }$
Then:
{{begin-eqn}}
{{eqn | l = x
| r = 1 + \frac 1 x
| c = substituting for $x$
}}
{{eqn | ll= \leadsto
| l = x^2
| r = x + 1
| c =
}}
{{eqn | ll= \leadsto
| l = x^2 - x - 1
| r = 0
| c =
}}
{{end-eqn}}... | Continued Fraction Expansion of Golden Mean | https://proofwiki.org/wiki/Continued_Fraction_Expansion_of_Golden_Mean | https://proofwiki.org/wiki/Continued_Fraction_Expansion_of_Golden_Mean | [
"Continued Fraction Expansion of Golden Mean",
"Golden Mean",
"Fibonacci Numbers",
"Examples of Continued Fractions"
] | [
"Definition:Golden Mean",
"Definition:Continued Fraction Expansion/Real Number"
] | [
"Golden Mean as Root of Quadratic"
] |
proofwiki-11518 | Fibonacci Number less than Golden Section to Power less One | For all $n \in \N_{> 0}$:
:$F_n \le \phi^{n - 1}$
where:
:$F_n$ is the $n$th Fibonacci number
:$\phi$ is the golden section: $\phi = \dfrac {1 + \sqrt 5} 2$ | The proof proceeds by induction.
For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:
:$F_n \le \phi^{n - 1}$ | For all $n \in \N_{> 0}$:
:$F_n \le \phi^{n - 1}$
where:
:$F_n$ is the $n$th [[Definition:Fibonacci Numbers|Fibonacci number]]
:$\phi$ is the [[Definition:Golden Section|golden section]]: $\phi = \dfrac {1 + \sqrt 5} 2$ | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \N_{> 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$F_n \le \phi^{n - 1}$ | Fibonacci Number less than Golden Section to Power less One | https://proofwiki.org/wiki/Fibonacci_Number_less_than_Golden_Section_to_Power_less_One | https://proofwiki.org/wiki/Fibonacci_Number_less_than_Golden_Section_to_Power_less_One | [
"Fibonacci Numbers",
"Golden Mean"
] | [
"Definition:Fibonacci Number",
"Definition:Golden Mean"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition"
] |
proofwiki-11519 | Square of Golden Mean equals One plus Golden Mean | :$\phi^2 = \phi + 1$
where $\phi$ denotes the golden mean. | {{begin-eqn}}
{{eqn | l = \phi
| r = \frac 1 {\phi - 1}
| c = {{Defof|Golden Mean|index = 3}}
}}
{{eqn | ll= \leadstoandfrom
| l = \phi \paren {\phi - 1}
| r = 1
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \phi^2
| r = \phi + 1
| c =
}}
{{end-eqn}}
{{qed}} | :$\phi^2 = \phi + 1$
where $\phi$ denotes the [[Definition:Golden Mean|golden mean]]. | {{begin-eqn}}
{{eqn | l = \phi
| r = \frac 1 {\phi - 1}
| c = {{Defof|Golden Mean|index = 3}}
}}
{{eqn | ll= \leadstoandfrom
| l = \phi \paren {\phi - 1}
| r = 1
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \phi^2
| r = \phi + 1
| c =
}}
{{end-eqn}}
{{qed}} | Square of Golden Mean equals One plus Golden Mean | https://proofwiki.org/wiki/Square_of_Golden_Mean_equals_One_plus_Golden_Mean | https://proofwiki.org/wiki/Square_of_Golden_Mean_equals_One_plus_Golden_Mean | [
"Golden Mean"
] | [
"Definition:Golden Mean"
] | [] |
proofwiki-11520 | Fibonacci Number greater than Golden Section to Power less Two | For all $n \in \N_{\ge 2}$:
:$F_n \ge \phi^{n - 2}$
where:
:$F_n$ is the $n$th Fibonacci number
:$\phi$ is the golden section: $\phi = \dfrac {1 + \sqrt 5} 2$ | The proof proceeds by induction.
For all $n \in \N_{\ge 2}$, let $\map P n$ be the proposition:
:$F_n \ge \phi^{n - 2}$ | For all $n \in \N_{\ge 2}$:
:$F_n \ge \phi^{n - 2}$
where:
:$F_n$ is the $n$th [[Definition:Fibonacci Numbers|Fibonacci number]]
:$\phi$ is the [[Definition:Golden Section|golden section]]: $\phi = \dfrac {1 + \sqrt 5} 2$ | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \N_{\ge 2}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$F_n \ge \phi^{n - 2}$ | Fibonacci Number greater than Golden Section to Power less Two | https://proofwiki.org/wiki/Fibonacci_Number_greater_than_Golden_Section_to_Power_less_Two | https://proofwiki.org/wiki/Fibonacci_Number_greater_than_Golden_Section_to_Power_less_Two | [
"Fibonacci Numbers",
"Golden Mean"
] | [
"Definition:Fibonacci Number",
"Definition:Golden Mean"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition"
] |
proofwiki-11521 | Triangular Number as Alternating Sum and Difference of Squares | {{begin-eqn}}
{{eqn | q = \forall n \in \N
| l = \frac {n \paren {n + 1} } 2
| r = \sum_{j \mathop = 0}^{n - 1} \paren {-1}^j \paren {n - j}^2
| c =
}}
{{eqn | r = n^2 - \paren {n - 1}^2 + \paren {n - 2}^2 - \cdots + \paren {-1}^{n - 1}
| c =
}}
{{end-eqn}}
Thus the $n$th triangular number ca... | The proof proceeds by induction.
For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:
:$\ds \frac {n \paren {n + 1} } 2 = \sum_{j \mathop = 0}^{n - 1} \paren {-1}^j \paren {n - j}^2$ | {{begin-eqn}}
{{eqn | q = \forall n \in \N
| l = \frac {n \paren {n + 1} } 2
| r = \sum_{j \mathop = 0}^{n - 1} \paren {-1}^j \paren {n - j}^2
| c =
}}
{{eqn | r = n^2 - \paren {n - 1}^2 + \paren {n - 2}^2 - \cdots + \paren {-1}^{n - 1}
| c =
}}
{{end-eqn}}
Thus the $n$th [[Definition:Triang... | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \N_{> 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \frac {n \paren {n + 1} } 2 = \sum_{j \mathop = 0}^{n - 1} \paren {-1}^j \paren {n - j}^2$ | Triangular Number as Alternating Sum and Difference of Squares | https://proofwiki.org/wiki/Triangular_Number_as_Alternating_Sum_and_Difference_of_Squares | https://proofwiki.org/wiki/Triangular_Number_as_Alternating_Sum_and_Difference_of_Squares | [
"Triangular Numbers",
"Square Numbers"
] | [
"Definition:Triangular Number",
"Definition:Square Number"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-11522 | Bernoulli's Inequality/Corollary | Let $x \in \R$ be a real number such that $0 < x < 1$.
Let $n \in \Z_{\ge 0}$ be a positive integer.
Then:
:$\paren {1 - x}^n \ge 1 - n x$ | Let $0 < x < 1$.
Let $y = -x$.
Then $y > -1$ and by Bernoulli's Inequality:
:$\paren {1 + y}^n \ge 1 + n y$
Thus:
:$\paren {1 + \paren {-x} }^n \ge 1 + n \paren {-x}$
Hence the result.
{{qed}} | Let $x \in \R$ be a [[Definition:Real Number|real number]] such that $0 < x < 1$.
Let $n \in \Z_{\ge 0}$ be a [[Definition:Positive Integer|positive integer]].
Then:
:$\paren {1 - x}^n \ge 1 - n x$ | Let $0 < x < 1$.
Let $y = -x$.
Then $y > -1$ and by [[Bernoulli's Inequality]]:
:$\paren {1 + y}^n \ge 1 + n y$
Thus:
:$\paren {1 + \paren {-x} }^n \ge 1 + n \paren {-x}$
Hence the result.
{{qed}} | Bernoulli's Inequality/Corollary/Proof 1 | https://proofwiki.org/wiki/Bernoulli's_Inequality/Corollary | https://proofwiki.org/wiki/Bernoulli's_Inequality/Corollary/Proof_1 | [
"Real Analysis",
"Inequalities",
"Bernoulli's Inequality"
] | [
"Definition:Real Number",
"Definition:Positive/Integer"
] | [
"Bernoulli's Inequality"
] |
proofwiki-11523 | Bernoulli's Inequality/Corollary | Let $x \in \R$ be a real number such that $0 < x < 1$.
Let $n \in \Z_{\ge 0}$ be a positive integer.
Then:
:$\paren {1 - x}^n \ge 1 - n x$ | Proof by induction:
Let $0 < x < 1$.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$\paren {1 - x}^n \ge 1 - n x$
=== Basis for the Induction ===
$\map P 0$ is the case:
:$\paren {1 - x}^0 = 1 \ge 1 - 0 x = 1$
so $\map P 0$ holds.
This is our basis for the induction.
=== Induction Hypothesis ===
Now ... | Let $x \in \R$ be a [[Definition:Real Number|real number]] such that $0 < x < 1$.
Let $n \in \Z_{\ge 0}$ be a [[Definition:Positive Integer|positive integer]].
Then:
:$\paren {1 - x}^n \ge 1 - n x$ | Proof by [[Principle of Mathematical Induction|induction]]:
Let $0 < x < 1$.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\paren {1 - x}^n \ge 1 - n x$
=== Basis for the Induction ===
$\map P 0$ is the case:
:$\paren {1 - x}^0 = 1 \ge 1 - 0 x = 1$
so $\map P 0$ holds... | Bernoulli's Inequality/Corollary/Proof 2 | https://proofwiki.org/wiki/Bernoulli's_Inequality/Corollary | https://proofwiki.org/wiki/Bernoulli's_Inequality/Corollary/Proof_2 | [
"Real Analysis",
"Inequalities",
"Bernoulli's Inequality"
] | [
"Definition:Real Number",
"Definition:Positive/Integer"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Bernoulli's Inequality/Corollary/Proof 2",
"Principle of Mathematical Induction"
] |
proofwiki-11524 | 2 to the n is Greater than n Cubed when n is 10 and above | :$\forall n \in \Z, n \ge 10: 2^n > n^3$ | Proof by induction:
For all $n \in \Z$ such that $n \ge 10$, let $\map P n$ be the proposition:
:$2^n > n^3$
We note that:
:$2^9 = 512 < 729 = 9^3$
so when $n < 10$ the proposition does not hold. | :$\forall n \in \Z, n \ge 10: 2^n > n^3$ | Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \Z$ such that $n \ge 10$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$2^n > n^3$
We note that:
:$2^9 = 512 < 729 = 9^3$
so when $n < 10$ the [[Definition:Proposition|proposition]] does not hold. | 2 to the n is Greater than n Cubed when n is 10 and above | https://proofwiki.org/wiki/2_to_the_n_is_Greater_than_n_Cubed_when_n_is_10_and_above | https://proofwiki.org/wiki/2_to_the_n_is_Greater_than_n_Cubed_when_n_is_10_and_above | [
"Number Theory",
"2",
"10"
] | [] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-11525 | Set of Integers is not Well-Ordered by Usual Ordering | The set of integers $\Z$ is not well-ordered under the usual ordering $\le$. | {{AimForCont}} $\Z$ is a well-ordered set.
Then by definition, all subsets of $\Z$ has a smallest element.
But take $\Z$ itself.
Suppose $x \in \Z$ is a smallest element.
Then $x - 1 \in \Z$.
But $x - 1 < x$, which contradicts the supposition that $x \in \Z$ is a smallest element.
Hence there can be no such smallest el... | The [[Definition:Integer|set of integers]] $\Z$ is not [[Definition:Well-Ordered Set|well-ordered]] under the [[Definition:Usual Ordering|usual ordering]] $\le$. | {{AimForCont}} $\Z$ is a [[Definition:Well-Ordered Set|well-ordered set]].
Then by definition, all [[Definition:Subset|subsets]] of $\Z$ has a [[Definition:Smallest Element|smallest element]].
But take $\Z$ itself.
Suppose $x \in \Z$ is a [[Definition:Smallest Element|smallest element]].
Then $x - 1 \in \Z$.
But $... | Set of Integers is not Well-Ordered by Usual Ordering | https://proofwiki.org/wiki/Set_of_Integers_is_not_Well-Ordered_by_Usual_Ordering | https://proofwiki.org/wiki/Set_of_Integers_is_not_Well-Ordered_by_Usual_Ordering | [
"Integers",
"Well-Orderings"
] | [
"Definition:Integer",
"Definition:Well-Ordered Set",
"Definition:Usual Ordering"
] | [
"Definition:Well-Ordered Set",
"Definition:Subset",
"Definition:Smallest Element",
"Definition:Smallest Element",
"Definition:Contradiction",
"Definition:Smallest Element",
"Definition:Smallest Element",
"Proof by Contradiction",
"Definition:Well-Ordered Set"
] |
proofwiki-11526 | Set of Integers can be Well-Ordered | The set of integers $\Z$ can be well-ordered with an appropriately chosen ordering. | Consider the ordering $\preccurlyeq \subseteq \Z \times \Z$ defined as:
:$x \preccurlyeq y \iff \left({\left\vert{x}\right\vert < \left\vert{y}\right\vert}\right) \lor \left({\left\vert{x}\right\vert = \left\vert{y}\right\vert \land x \le y}\right)$
{{finish|It remains to be shown that $\preccurlyeq$ is an ordering, an... | The [[Definition:Integer|set of integers]] $\Z$ can be [[Definition:Well-Ordered Set|well-ordered]] with an appropriately chosen [[Definition:Ordering|ordering]]. | Consider the [[Definition:Ordering|ordering]] $\preccurlyeq \subseteq \Z \times \Z$ defined as:
:$x \preccurlyeq y \iff \left({\left\vert{x}\right\vert < \left\vert{y}\right\vert}\right) \lor \left({\left\vert{x}\right\vert = \left\vert{y}\right\vert \land x \le y}\right)$
{{finish|It remains to be shown that $\preccu... | Set of Integers can be Well-Ordered | https://proofwiki.org/wiki/Set_of_Integers_can_be_Well-Ordered | https://proofwiki.org/wiki/Set_of_Integers_can_be_Well-Ordered | [
"Integers",
"Well-Orderings"
] | [
"Definition:Integer",
"Definition:Well-Ordered Set",
"Definition:Ordering"
] | [
"Definition:Ordering"
] |
proofwiki-11527 | Set of Non-Negative Real Numbers is not Well-Ordered by Usual Ordering | The set of non-negative real numbers $\R_{\ge 0}$ is not well-ordered under the usual ordering $\le$. | {{AimForCont}} $\R_{\ge 0}$ is a well-ordered set.
Then by definition, all subsets of $\R_{\ge 0}$ has a smallest element.
Take the subset $\R_{> 0}$:
:$\R_{> 0} = \left\{ {x \in \R_{\ge 0}: x > 0}\right\} = \R_{\ge 0} \setminus \left\{ {0}\right\}$
Suppose $x \in \R_{> 0}$ is a smallest element.
Then $\dfrac x 2 \in \... | The [[Definition:Positive Real Number|set of non-negative real numbers]] $\R_{\ge 0}$ is not [[Definition:Well-Ordered Set|well-ordered]] under the [[Definition:Usual Ordering|usual ordering]] $\le$. | {{AimForCont}} $\R_{\ge 0}$ is a [[Definition:Well-Ordered Set|well-ordered set]].
Then by definition, all [[Definition:Subset|subsets]] of $\R_{\ge 0}$ has a [[Definition:Smallest Element|smallest element]].
Take the subset $\R_{> 0}$:
:$\R_{> 0} = \left\{ {x \in \R_{\ge 0}: x > 0}\right\} = \R_{\ge 0} \setminus \le... | Set of Non-Negative Real Numbers is not Well-Ordered by Usual Ordering | https://proofwiki.org/wiki/Set_of_Non-Negative_Real_Numbers_is_not_Well-Ordered_by_Usual_Ordering | https://proofwiki.org/wiki/Set_of_Non-Negative_Real_Numbers_is_not_Well-Ordered_by_Usual_Ordering | [
"Real Numbers",
"Well-Orderings"
] | [
"Definition:Positive/Real Number",
"Definition:Well-Ordered Set",
"Definition:Usual Ordering"
] | [
"Definition:Well-Ordered Set",
"Definition:Subset",
"Definition:Smallest Element",
"Definition:Smallest Element",
"Definition:Contradiction",
"Definition:Smallest Element",
"Definition:Smallest Element",
"Proof by Contradiction",
"Definition:Well-Ordered Set"
] |
proofwiki-11528 | Existence of Digital Root | Let $n \in \N$ be a natural number.
Let $b \in \N$ such that $b \ge 2$ also be a natural number.
Let $n$ be expressed in base $b$.
Then the digital root base $b$ exists for $n$. | By definition, the digital root base $b$ for $n$ is the single digit resulting from:
:adding up the digits in $n$, and expressing the result in base $b$
:adding up the digits in that result, and again expressing the result in base $b$
:repeating until down to one digit.
Let $n = d_1 + b d_2 + \dotsb + b^{m - 1} d_m$ wh... | Let $n \in \N$ be a [[Definition:Natural Number|natural number]].
Let $b \in \N$ such that $b \ge 2$ also be a [[Definition:Natural Number|natural number]].
Let $n$ be expressed in [[Definition:Number Base|base $b$]].
Then the [[Definition:Digital Root|digital root]] [[Definition:Number Base|base $b$]] exists for $... | By definition, the [[Definition:Digital Root|digital root]] [[Definition:Number Base|base $b$]] for $n$ is the single [[Definition:Digit|digit]] resulting from:
:adding up the [[Definition:Digit|digits]] in $n$, and expressing the result in [[Definition:Number Base|base $b$]]
:adding up the [[Definition:Digit|digits]] ... | Existence of Digital Root | https://proofwiki.org/wiki/Existence_of_Digital_Root | https://proofwiki.org/wiki/Existence_of_Digital_Root | [
"Number Bases",
"Digital Roots"
] | [
"Definition:Natural Numbers",
"Definition:Natural Numbers",
"Definition:Number Base",
"Definition:Digital Root",
"Definition:Number Base"
] | [
"Definition:Digital Root",
"Definition:Number Base",
"Definition:Digit",
"Definition:Digit",
"Definition:Number Base",
"Definition:Digit",
"Definition:Number Base",
"Definition:Digit",
"Definition:Digit Sum",
"Definition:Digit",
"Definition:Digit Sum",
"Definition:Digit Sum",
"Definition:Fin... |
proofwiki-11529 | Golden Mean is Irrational | The golden mean $\phi$ is irrational. | By definition of golden mean:
:$\phi = \dfrac {1 + \sqrt 5} 2$
By Square Root of Prime is Irrational:
:$\sqrt 5$ is irrational.
By Rational Number plus Irrational Number is Irrational:
:$1 + \sqrt 5$ is irrational.
By Irrational Number divided by Rational Number is Irrational:
:$\dfrac {1 + \sqrt 5} 2$ is irrational.
{... | The [[Definition:Golden Mean|golden mean]] $\phi$ is [[Definition:Irrational Number|irrational]]. | By definition of [[Definition:Golden Mean|golden mean]]:
:$\phi = \dfrac {1 + \sqrt 5} 2$
By [[Square Root of Prime is Irrational]]:
:$\sqrt 5$ is [[Definition:Irrational Number|irrational]].
By [[Rational Number plus Irrational Number is Irrational]]:
:$1 + \sqrt 5$ is [[Definition:Irrational Number|irrational]].
B... | Golden Mean is Irrational | https://proofwiki.org/wiki/Golden_Mean_is_Irrational | https://proofwiki.org/wiki/Golden_Mean_is_Irrational | [
"Golden Mean"
] | [
"Definition:Golden Mean",
"Definition:Irrational Number"
] | [
"Definition:Golden Mean",
"Square Root of Prime is Irrational",
"Definition:Irrational Number",
"Rational Number plus Irrational Number is Irrational",
"Definition:Irrational Number",
"Irrational Number divided by Rational Number is Irrational",
"Definition:Irrational Number"
] |
proofwiki-11530 | Way Below iff Preceding Finite Supremum | Let $\struct {S, \vee, \wedge, \preceq}$ be a complete lattice.
Let $x, y \in S$.
Then $x \ll y$ {{iff}}
:$\forall X \subseteq S: y \preceq \sup X \implies \exists A \in \map {\it Fin} X: x \preceq \sup A$
where
:$\ll$ denotes the way below relation,
:$\map {\it Fin} X$ denotes the set of all finite subsets of $X$. | === Sufficient Condition ===
Let $x \ll y$
Let $X \subseteq S$ such that:
:$y \preceq \sup X$
Define:
:$F := \set {\sup A: A \in \map {\it Fin} X}$
By definition of union:
:$X = \bigcup \map {\it Fin} X$
By Supremum of Suprema:
:$\sup X = \sup F$
We will prove that:
:$F$ is directed
Let $a, b \in F$.
By definition of $... | Let $\struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Complete Lattice|complete lattice]].
Let $x, y \in S$.
Then $x \ll y$ {{iff}}
:$\forall X \subseteq S: y \preceq \sup X \implies \exists A \in \map {\it Fin} X: x \preceq \sup A$
where
:$\ll$ denotes the [[Definition:Element is Way Below|way below relation]]... | === Sufficient Condition ===
Let $x \ll y$
Let $X \subseteq S$ such that:
:$y \preceq \sup X$
Define:
:$F := \set {\sup A: A \in \map {\it Fin} X}$
By definition of [[Definition:Set Union|union]]:
:$X = \bigcup \map {\it Fin} X$
By [[Supremum of Suprema]]:
:$\sup X = \sup F$
We will prove that:
:$F$ is [[Definiti... | Way Below iff Preceding Finite Supremum | https://proofwiki.org/wiki/Way_Below_iff_Preceding_Finite_Supremum | https://proofwiki.org/wiki/Way_Below_iff_Preceding_Finite_Supremum | [
"Way Below Relation"
] | [
"Definition:Complete Lattice",
"Definition:Element is Way Below",
"Definition:Set of Sets",
"Definition:Finite Set",
"Definition:Subset"
] | [
"Definition:Set Union",
"Supremum of Suprema",
"Definition:Directed Subset",
"Union of Subsets is Subset",
"Finite Union of Finite Sets is Finite",
"Definition:Finite Set",
"Set is Subset of Union",
"Supremum of Subset",
"Definition:Directed Subset",
"Definition:Element is Way Below",
"Definitio... |
proofwiki-11531 | Exponent Combination Laws/Product of Powers/Proof 2 | Let $a \in \R_{> 0}$ be a positive real number.
Let $x, y \in \R$ be real numbers.
Let $a^x$ be defined as $a$ to the power of $x$.
Then:
:$a^x a^y = a^{x + y}$ | Let $x, y \in \R$.
From Rational Sequence Decreasing to Real Number, there exist rational sequences $\sequence {x_n}$ and $\sequence {y_n}$ converging to $x$ and $y$, respectively.
Then, since Power Function on Strictly Positive Base is Continuous: Real Power:
{{begin-eqn}}
{{eqn | l = a^{x + y}
| r = a^{\ds \par... | Let $a \in \R_{> 0}$ be a [[Definition:Positive Real Number|positive real number]].
Let $x, y \in \R$ be [[Definition:Real Number|real numbers]].
Let $a^x$ be defined as [[Definition:Power to Real Number|$a$ to the power of $x$]].
Then:
:$a^x a^y = a^{x + y}$ | Let $x, y \in \R$.
From [[Rational Sequence Decreasing to Real Number]], there exist [[Definition:Rational Sequence|rational sequences]] $\sequence {x_n}$ and $\sequence {y_n}$ [[Definition:Convergent Real Sequence|converging]] to $x$ and $y$, respectively.
Then, since [[Power Function on Strictly Positive Base is ... | Exponent Combination Laws/Product of Powers/Proof 2 | https://proofwiki.org/wiki/Exponent_Combination_Laws/Product_of_Powers/Proof_2 | https://proofwiki.org/wiki/Exponent_Combination_Laws/Product_of_Powers/Proof_2 | [
"Exponent Combination Laws"
] | [
"Definition:Positive/Real Number",
"Definition:Real Number",
"Definition:Power (Algebra)/Real Number"
] | [
"Rational Sequence Decreasing to Real Number",
"Definition:Rational Sequence",
"Definition:Convergent Sequence/Real Numbers",
"Power Function on Strictly Positive Base is Continuous/Real Power",
"Combination Theorem for Sequences/Real/Sum Rule",
"Sequential Continuity is Equivalent to Continuity in the Re... |
proofwiki-11532 | Exponent Combination Laws/Power of Power/Proof 1 | Let $a \in \R_{>0}$ be a (strictly) positive real number.
{{:Power of Power}} | {{begin-eqn}}
{{eqn | l = a^{x y}
| r = \map \exp {x y \ln a}
| c = {{Defof|Power to Real Number}}
}}
{{eqn | r = \map \exp {y \, \map \ln {a^x} }
| c = Logarithms of Powers
}}
{{eqn | r = \paren {a^x}^y
| c = {{Defof|Power to Real Number}}
}}
{{end-eqn}}
{{qed}} | Let $a \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|(strictly) positive real number]].
{{:Power of Power}} | {{begin-eqn}}
{{eqn | l = a^{x y}
| r = \map \exp {x y \ln a}
| c = {{Defof|Power to Real Number}}
}}
{{eqn | r = \map \exp {y \, \map \ln {a^x} }
| c = [[Logarithms of Powers]]
}}
{{eqn | r = \paren {a^x}^y
| c = {{Defof|Power to Real Number}}
}}
{{end-eqn}}
{{qed}} | Exponent Combination Laws/Power of Power/Proof 1 | https://proofwiki.org/wiki/Exponent_Combination_Laws/Power_of_Power/Proof_1 | https://proofwiki.org/wiki/Exponent_Combination_Laws/Power_of_Power/Proof_1 | [
"Exponent Combination Laws"
] | [
"Definition:Strictly Positive/Real Number"
] | [
"Logarithm of Power"
] |
proofwiki-11533 | Exponent Combination Laws/Power of Power/Proof 2 | Let $a \in \R_{>0}$ be a (strictly) positive real number.
{{:Power of Power}} | We will show that:
:$\forall \epsilon \in \R_{>0}: \size {a^{x y} - \paren {a^x}^y} < \epsilon$
{{WLOG}}, suppose that $x < y$.
Consider $I := \closedint x y$.
Let $I_\Q = I \cap \Q$.
Let $M = \max \set {\size x, \size y}$
Fix $\epsilon \in \R_{>0}$.
From Real Polynomial Function is Continuous:
:$\exists \delta' \in \R... | Let $a \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|(strictly) positive real number]].
{{:Power of Power}} | We will show that:
:$\forall \epsilon \in \R_{>0}: \size {a^{x y} - \paren {a^x}^y} < \epsilon$
{{WLOG}}, suppose that $x < y$.
Consider $I := \closedint x y$.
Let $I_\Q = I \cap \Q$.
Let $M = \max \set {\size x, \size y}$
Fix $\epsilon \in \R_{>0}$.
From [[Real Polynomial Function is Continuous]]:
:$\exists \del... | Exponent Combination Laws/Power of Power/Proof 2 | https://proofwiki.org/wiki/Exponent_Combination_Laws/Power_of_Power/Proof_2 | https://proofwiki.org/wiki/Exponent_Combination_Laws/Power_of_Power/Proof_2 | [
"Exponent Combination Laws"
] | [
"Definition:Strictly Positive/Real Number"
] | [
"Real Polynomial Function is Continuous",
"Power Function on Strictly Positive Base is Continuous",
"Absolute Value Function is Completely Multiplicative",
"Absolute Value Function is Completely Multiplicative",
"Real Number Ordering is Compatible with Multiplication",
"Closure of Rational Interval is Clo... |
proofwiki-11534 | Rational Sequence Increasing to Real Number | Let $x \in \R$ be a real number.
Then there exists some increasing rational sequence that converges to $x$. | Let $\sequence {x_n}$ denote the sequence defined as:
:$\forall n \in \N: x_n = \dfrac {\floor {n x} } n$
where $\floor {n x}$ denotes the floor of $n x$.
From Floor Function is Integer, $\floor {n x}$ is an integer.
Hence by definition of rational number, $\sequence {x_n}$ is a rational sequence.
From Real Number is b... | Let $x \in \R$ be a [[Definition:Real Number|real number]].
Then there exists some [[Definition:Increasing Sequence|increasing]] [[Definition:Rational Sequence|rational sequence]] that [[Definition:Convergent Real Sequence|converges]] to $x$. | Let $\sequence {x_n}$ denote the [[Definition:Sequence|sequence]] defined as:
:$\forall n \in \N: x_n = \dfrac {\floor {n x} } n$
where $\floor {n x}$ denotes the [[Definition:Floor Function|floor]] of $n x$.
From [[Floor Function is Integer]], $\floor {n x}$ is an [[Definition:Integer|integer]].
Hence by definition... | Rational Sequence Increasing to Real Number | https://proofwiki.org/wiki/Rational_Sequence_Increasing_to_Real_Number | https://proofwiki.org/wiki/Rational_Sequence_Increasing_to_Real_Number | [
"Limits of Sequences",
"Rational Sequences"
] | [
"Definition:Real Number",
"Definition:Increasing/Sequence",
"Definition:Rational Sequence",
"Definition:Convergent Sequence/Real Numbers"
] | [
"Definition:Sequence",
"Definition:Floor Function",
"Floor Function is Integer",
"Definition:Integer",
"Definition:Rational Number",
"Definition:Rational Sequence",
"Real Number is between Floor Functions",
"Combination Theorem for Sequences/Real/Combined Sum Rule",
"Combination Theorem for Limits o... |
proofwiki-11535 | Rational Sequence Decreasing to Real Number | Let $x \in \R$ be a real number.
Then there exists some decreasing rational sequence that converges to $x$. | Let $\sequence {x_n}$ denote the sequence defined as:
:$\forall n \in \N : x_n = \dfrac {\ceiling {n x} } n$
where $\ceiling {n x}$ denotes the ceiling of $n x$.
From Ceiling Function is Integer, $\ceiling {n x}$ is an integer.
Hence by definition of rational number, $\sequence {x_n}$ is a rational sequence.
From Real ... | Let $x \in \R$ be a [[Definition:Real Number|real number]].
Then there exists some [[Definition:Decreasing Sequence|decreasing]] [[Definition:Rational Sequence|rational sequence]] that [[Definition:Convergent Real Sequence|converges]] to $x$. | Let $\sequence {x_n}$ denote the [[Definition:Real Sequence|sequence]] defined as:
:$\forall n \in \N : x_n = \dfrac {\ceiling {n x} } n$
where $\ceiling {n x}$ denotes the [[Definition:Ceiling Function|ceiling]] of $n x$.
From [[Ceiling Function is Integer]], $\ceiling {n x}$ is an [[Definition:Integer|integer]].
H... | Rational Sequence Decreasing to Real Number | https://proofwiki.org/wiki/Rational_Sequence_Decreasing_to_Real_Number | https://proofwiki.org/wiki/Rational_Sequence_Decreasing_to_Real_Number | [
"Limits of Sequences",
"Rational Sequences"
] | [
"Definition:Real Number",
"Definition:Decreasing/Sequence",
"Definition:Rational Sequence",
"Definition:Convergent Sequence/Real Numbers"
] | [
"Definition:Real Sequence",
"Definition:Ceiling Function",
"Ceiling Function is Integer",
"Definition:Integer",
"Definition:Rational Number",
"Definition:Rational Sequence",
"Real Number is between Ceiling Functions",
"Combination Theorem for Sequences/Real/Combined Sum Rule",
"Combination Theorem f... |
proofwiki-11536 | Power Function on Base Greater than One is Strictly Increasing/Real Number | Let $a \in \R$ be a real number such that $a > 1$.
Let $f: \R \to \R$ be the real function defined as:
:$\map f x = a^x$
where $a^x$ denotes $a$ to the power of $x$.
Then $f$ is strictly increasing. | Let $x, y \in \R$ be such that $x < y$.
Let $\delta = \dfrac {y - x} 2$.
From Rational Sequence Decreasing to Real Number, there is some rational sequence $\sequence {x_n}$ that decreases to $x$.
From Rational Sequence Increasing to Real Number, there is some rational sequence $\sequence {y_n}$ that increases to $y$.
F... | Let $a \in \R$ be a [[Definition:Real Number|real number]] such that $a > 1$.
Let $f: \R \to \R$ be the [[Definition:Real Function|real function]] defined as:
:$\map f x = a^x$
where $a^x$ denotes [[Definition:Power to Real Number|$a$ to the power of $x$]].
Then $f$ is [[Definition:Strictly Increasing Real Function|... | Let $x, y \in \R$ be such that $x < y$.
Let $\delta = \dfrac {y - x} 2$.
From [[Rational Sequence Decreasing to Real Number]], there is some [[Definition:Rational Sequence|rational sequence]] $\sequence {x_n}$ that [[Definition:Decreasing Sequence|decreases]] to $x$.
From [[Rational Sequence Increasing to Real Numb... | Power Function on Base Greater than One is Strictly Increasing/Real Number | https://proofwiki.org/wiki/Power_Function_on_Base_Greater_than_One_is_Strictly_Increasing/Real_Number | https://proofwiki.org/wiki/Power_Function_on_Base_Greater_than_One_is_Strictly_Increasing/Real_Number | [
"Power Function on Base Greater than One is Strictly Increasing",
"Real Analysis"
] | [
"Definition:Real Number",
"Definition:Real Function",
"Definition:Power (Algebra)/Real Number",
"Definition:Strictly Increasing/Real Function"
] | [
"Rational Sequence Decreasing to Real Number",
"Definition:Rational Sequence",
"Definition:Decreasing/Sequence",
"Rational Sequence Increasing to Real Number",
"Definition:Rational Sequence",
"Definition:Increasing/Sequence",
"Convergent Real Sequence is Bounded",
"Definition:Decreasing/Sequence",
"... |
proofwiki-11537 | Power Function on Strictly Positive Base is Continuous/Real Power | Let $f : \R \to \R$ be the real function defined as:
:$\map f x = a^x$
where $a^x$ denotes $a$ to the power of $x$.
Then $f$ is continuous. | By definition, $a^x$ is the unique continuous extension of $a^r$, for rational $r$.
By definition, continuous extensions are continuous.
Hence the result.
{{qed}} | Let $f : \R \to \R$ be the [[Definition:Real Function|real function]] defined as:
:$\map f x = a^x$
where $a^x$ denotes [[Definition:Power (Algebra)/Real Number/Definition 2|$a$ to the power of $x$]].
Then $f$ is [[Definition:Continuous Real Function|continuous]]. | By definition, $a^x$ is the [[Definition:Unique|unique]] [[Definition:Real Continuous Extension|continuous extension]] of $a^r$, for [[Definition:Rational Number|rational]] $r$.
By definition, [[Definition:Real Continuous Extension|continuous extensions]] are [[Definition:Continuous Real Function|continuous]].
Hence... | Power Function on Strictly Positive Base is Continuous/Real Power | https://proofwiki.org/wiki/Power_Function_on_Strictly_Positive_Base_is_Continuous/Real_Power | https://proofwiki.org/wiki/Power_Function_on_Strictly_Positive_Base_is_Continuous/Real_Power | [
"Powers"
] | [
"Definition:Real Function",
"Definition:Power (Algebra)/Real Number/Definition 2",
"Definition:Continuous Real Function"
] | [
"Definition:Unique",
"Definition:Continuous Extension/Real Function",
"Definition:Rational Number",
"Definition:Continuous Extension/Real Function",
"Definition:Continuous Real Function"
] |
proofwiki-11538 | Power Function on Strictly Positive Base is Continuous/Rational Power | Let $f: \Q \to \R$ be the real-valued function defined as:
:$\map f x = a^x$
where $a^x$ denotes $a$ to the power of $x$.
Then $f$ is continuous. | {{proof wanted|first part of proof of Power Function to Rational Power permits Unique Continuous Extension}}
Category:Powers
9cv5bkcj132536wk0xqjpjveimowda4 | Let $f: \Q \to \R$ be the [[Definition:Real-Valued Function|real-valued function]] defined as:
:$\map f x = a^x$
where $a^x$ denotes [[Definition:Rational Power|$a$ to the power of $x$]].
Then $f$ is [[Definition:Continuous Real Function|continuous]]. | {{proof wanted|first part of proof of [[Power Function to Rational Power permits Unique Continuous Extension]]}}
[[Category:Powers]]
9cv5bkcj132536wk0xqjpjveimowda4 | Power Function on Strictly Positive Base is Continuous/Rational Power | https://proofwiki.org/wiki/Power_Function_on_Strictly_Positive_Base_is_Continuous/Rational_Power | https://proofwiki.org/wiki/Power_Function_on_Strictly_Positive_Base_is_Continuous/Rational_Power | [
"Powers"
] | [
"Definition:Real-Valued Function",
"Definition:Power (Algebra)/Rational Number",
"Definition:Continuous Real Function"
] | [
"Power Function to Rational Power permits Unique Continuous Extension",
"Category:Powers"
] |
proofwiki-11539 | Defining Sequence of Natural Logarithm is Convergent | Let $x \in \R$ be a real number such that $x > 0$.
Let $\sequence {f_n}$ be the sequence of mappings $f_n : \R_{>0} \to \R$ defined as:
:$\map {f_n} x = n \paren {\sqrt[n] x - 1}$
Then $\sequence {f_n}$ is pointwise convergent. | Fix $x \in \R_{>0}$.
From Defining Sequence of Natural Logarithm is Strictly Decreasing, $\sequence {\map {f_n} x}$ is strictly decreasing.
From Lower Bound of Natural Logarithm, $\sequence {\map {f_n} x}$ is bounded below.
From Monotone Convergence Theorem, $\sequence {\map {f_n} x}$ is convergent.
Hence the result, b... | Let $x \in \R$ be a [[Definition:Real Number|real number]] such that $x > 0$.
Let $\sequence {f_n}$ be the [[Definition:Sequence|sequence]] of mappings $f_n : \R_{>0} \to \R$ defined as:
:$\map {f_n} x = n \paren {\sqrt[n] x - 1}$
Then $\sequence {f_n}$ is [[Definition:Pointwise Convergence|pointwise convergent]]. | Fix $x \in \R_{>0}$.
From [[Defining Sequence of Natural Logarithm is Strictly Decreasing]], $\sequence {\map {f_n} x}$ is [[Definition:Strictly Decreasing Sequence|strictly decreasing]].
From [[Lower Bound of Natural Logarithm]], $\sequence {\map {f_n} x}$ is [[Definition:Bounded Below Real Sequence|bounded below]].... | Defining Sequence of Natural Logarithm is Convergent | https://proofwiki.org/wiki/Defining_Sequence_of_Natural_Logarithm_is_Convergent | https://proofwiki.org/wiki/Defining_Sequence_of_Natural_Logarithm_is_Convergent | [
"Natural Logarithms"
] | [
"Definition:Real Number",
"Definition:Sequence",
"Definition:Pointwise Convergence"
] | [
"Defining Sequence of Natural Logarithm is Strictly Decreasing",
"Definition:Strictly Decreasing/Sequence",
"Lower Bound of Natural Logarithm",
"Definition:Bounded Below Sequence/Real",
"Monotone Convergence Theorem (Real Analysis)",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Pointwise C... |
proofwiki-11540 | Sum of Indices of Real Number/Rational Numbers | Let $x, y \in \Q$ be rational numbers.
Let $r^x$ be defined as $r$ to the power of $n$.
Then:
: $r^{x + y} = r^x \times r^y$ | Let $x = \dfrac p q, y = \dfrac u v$.
Then:
{{begin-eqn}}
{{eqn | l = r^\paren {x + y}
| r = r^\paren {\paren {p / q} + \paren {u / v} }
| c =
}}
{{eqn | r = r^\paren {\paren {p v + u q} / q v}
| c =
}}
{{eqn | r = \paren {r^\paren {1 / q v} }^\paren {p v + u q}
| c = {{Defof|Rational Power}}
... | Let $x, y \in \Q$ be [[Definition:Rational Number|rational numbers]].
Let $r^x$ be defined as [[Definition:Rational Power|$r$ to the power of $n$]].
Then:
: $r^{x + y} = r^x \times r^y$ | Let $x = \dfrac p q, y = \dfrac u v$.
Then:
{{begin-eqn}}
{{eqn | l = r^\paren {x + y}
| r = r^\paren {\paren {p / q} + \paren {u / v} }
| c =
}}
{{eqn | r = r^\paren {\paren {p v + u q} / q v}
| c =
}}
{{eqn | r = \paren {r^\paren {1 / q v} }^\paren {p v + u q}
| c = {{Defof|Rational Power}}... | Sum of Indices of Real Number/Rational Numbers | https://proofwiki.org/wiki/Sum_of_Indices_of_Real_Number/Rational_Numbers | https://proofwiki.org/wiki/Sum_of_Indices_of_Real_Number/Rational_Numbers | [
"Sum of Indices of Real Number"
] | [
"Definition:Rational Number",
"Definition:Power (Algebra)/Rational Number"
] | [
"Sum of Indices of Real Number/Integers"
] |
proofwiki-11541 | Product of Indices of Real Number/Rational Numbers | Let $x, y \in \Q$ be rational numbers.
Let $r^x$ be defined as $r$ to the power of $x$.
Then:
:$\paren {r^x}^y = r^{x y}$ | Let $x = \dfrac p q, y = \dfrac u v$.
Consider $\paren {\paren {r^x}^y}^{q v}$.
Then:
{{begin-eqn}}
{{eqn | l = \paren {\paren {r^x}^y}^{q v}
| r = \paren {\paren {r^\paren {p / q} }^\paren {u / v} }^{q v}
| c =
}}
{{eqn | r = \paren {\paren {\paren {r^\paren {p / q} }^\paren {u / v} }^v}^q
| c = Pro... | Let $x, y \in \Q$ be [[Definition:Rational Number|rational numbers]].
Let $r^x$ be defined as [[Definition:Rational Power|$r$ to the power of $x$]].
Then:
:$\paren {r^x}^y = r^{x y}$ | Let $x = \dfrac p q, y = \dfrac u v$.
Consider $\paren {\paren {r^x}^y}^{q v}$.
Then:
{{begin-eqn}}
{{eqn | l = \paren {\paren {r^x}^y}^{q v}
| r = \paren {\paren {r^\paren {p / q} }^\paren {u / v} }^{q v}
| c =
}}
{{eqn | r = \paren {\paren {\paren {r^\paren {p / q} }^\paren {u / v} }^v}^q
| c = [... | Product of Indices of Real Number/Rational Numbers | https://proofwiki.org/wiki/Product_of_Indices_of_Real_Number/Rational_Numbers | https://proofwiki.org/wiki/Product_of_Indices_of_Real_Number/Rational_Numbers | [
"Product of Indices of Real Number"
] | [
"Definition:Rational Number",
"Definition:Power (Algebra)/Rational Number"
] | [
"Product of Indices of Real Number/Integers",
"Product of Indices of Real Number/Integers",
"Product of Indices of Real Number/Integers",
"Product of Indices of Real Number/Integers",
"Definition:Root of Number"
] |
proofwiki-11542 | Sum of Indices of Real Number/Integers | Let $n, m \in \Z$ be integers.
Let $r^n$ be defined as $r$ to the power of $n$.
Then:
:$r^{n + m} = r^n \times r^m$ | From Sum of Indices of Real Number: Positive Integers, we have that:
:$m \in \Z_{\ge 0}: \forall n \in \Z: r^{n + m} = r^n \times r^m$
It remains to be shown that:
:$\forall m \in \Z_{<0}: \forall n \in \Z: r^{n + m} = r^n \times r^m$
The proof will proceed by induction on $m$.
As $m < 0$ we have that $m = -p$ for some... | Let $n, m \in \Z$ be [[Definition:Integer|integers]].
Let $r^n$ be defined as [[Definition:Integer Power|$r$ to the power of $n$]].
Then:
:$r^{n + m} = r^n \times r^m$ | From [[Sum of Indices of Real Number/Positive Integers|Sum of Indices of Real Number: Positive Integers]], we have that:
:$m \in \Z_{\ge 0}: \forall n \in \Z: r^{n + m} = r^n \times r^m$
It remains to be shown that:
:$\forall m \in \Z_{<0}: \forall n \in \Z: r^{n + m} = r^n \times r^m$
The proof will proceed by [... | Sum of Indices of Real Number/Integers | https://proofwiki.org/wiki/Sum_of_Indices_of_Real_Number/Integers | https://proofwiki.org/wiki/Sum_of_Indices_of_Real_Number/Integers | [
"Sum of Indices of Real Number"
] | [
"Definition:Integer",
"Definition:Power (Algebra)/Integer"
] | [
"Sum of Indices of Real Number/Positive Integers",
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-11543 | Product of Indices of Real Number/Integers | Let $n, m \in \Z$ be positive integers.
Let $r^n$ be defined as $r$ to the power of $n$.
Then:
:$\paren {r^n}^m = r^{n m}$ | From Product of Indices of Real Number: Positive Integers, we have that:
:$m \in \Z_{\ge 0}: \forall n \in \Z: \paren {r^n}^m = r^{n m}$
It remains to be shown that:
:$\forall m \in \Z_{<0}: \forall n \in \Z: \paren {r^n}^m = r^{n m}$
As $m < 0$ we have that $m = -p$ for some $p \in \Z_{> 0}$.
Thus:
{{begin-eqn}}
{{eqn... | Let $n, m \in \Z$ be [[Definition:Positive Integer|positive integers]].
Let $r^n$ be defined as [[Definition:Integer Power|$r$ to the power of $n$]].
Then:
:$\paren {r^n}^m = r^{n m}$ | From [[Product of Indices of Real Number/Positive Integers|Product of Indices of Real Number: Positive Integers]], we have that:
:$m \in \Z_{\ge 0}: \forall n \in \Z: \paren {r^n}^m = r^{n m}$
It remains to be shown that:
:$\forall m \in \Z_{<0}: \forall n \in \Z: \paren {r^n}^m = r^{n m}$
As $m < 0$ we have that... | Product of Indices of Real Number/Integers | https://proofwiki.org/wiki/Product_of_Indices_of_Real_Number/Integers | https://proofwiki.org/wiki/Product_of_Indices_of_Real_Number/Integers | [
"Product of Indices of Real Number"
] | [
"Definition:Positive/Integer",
"Definition:Power (Algebra)/Integer"
] | [
"Product of Indices of Real Number/Positive Integers",
"Real Number to Negative Power/Integer",
"Real Number to Negative Power/Integer"
] |
proofwiki-11544 | Uniformly Convergent iff Difference Under Supremum Metric Vanishes | Let $X$ be a set.
Let $\struct {Y, d}$ be a metric space.
Let $S$ be the set of all bounded mappings from $X$ to $Y$.
Let $\sequence {f_n}_{n \mathop \in \N}$ be a sequence in $S$.
Let $f \in S$.
Let $d_S: S \times S \to Y$ denote the supremum metric on $S$.
Then:
:$\sequence {f_n}$ converges uniformly to $f$ on $S$
{{... | {{begin-eqn}}
{{eqn | q = \forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N_{>N}: \forall x \in X
| l = \map d {\map {f_n} x, \map f x}
| o = <
| r = \epsilon
| c = {{Defof|Uniform Convergence}}
}}
{{eqn | ll= \leadstoandfrom
| q = \forall \epsilon \in \R_{>0}: \exists ... | Let $X$ be a [[Definition:Set|set]].
Let $\struct {Y, d}$ be a [[Definition:Metric Space|metric space]].
Let $S$ be the set of all [[Definition:Bounded Mapping to Metric Space|bounded mappings]] from $X$ to $Y$.
Let $\sequence {f_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] in $S$.
Let $f \in S$.
L... | {{begin-eqn}}
{{eqn | q = \forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N_{>N}: \forall x \in X
| l = \map d {\map {f_n} x, \map f x}
| o = <
| r = \epsilon
| c = {{Defof|Uniform Convergence}}
}}
{{eqn | ll= \leadstoandfrom
| q = \forall \epsilon \in \R_{>0}: \exists ... | Uniformly Convergent iff Difference Under Supremum Metric Vanishes | https://proofwiki.org/wiki/Uniformly_Convergent_iff_Difference_Under_Supremum_Metric_Vanishes | https://proofwiki.org/wiki/Uniformly_Convergent_iff_Difference_Under_Supremum_Metric_Vanishes | [
"Metric Spaces",
"Uniform Convergence"
] | [
"Definition:Set",
"Definition:Metric Space",
"Definition:Bounded Mapping/Metric Space",
"Definition:Sequence",
"Definition:Supremum Metric",
"Definition:Uniform Convergence"
] | [
"Category:Metric Spaces",
"Category:Uniform Convergence"
] |
proofwiki-11545 | Uniform Convergence is Hereditary | Let $M = \struct {A, d}$ be a metric space.
Let $\sequence {f_n}$ be a sequence of mappings defined on $A$.
Let $\sequence {f_n}$ be uniformly convergent on $S \subseteq A$.
Then $\sequence {f_n}$ is uniformly convergent on every metric subspace of $S$.
That is, uniform convergence is a hereditary property of a metric ... | {{ProofWanted}}
Category:Metric Spaces
3bkrpgwg5yp9kn5et5d2op3zfqsnlf6 | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $\sequence {f_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Mapping|mappings]] defined on $A$.
Let $\sequence {f_n}$ be [[Definition:Uniform Convergence|uniformly convergent]] on $S \subseteq A$.
Then $\sequence {f_n}$ is [[Defi... | {{ProofWanted}}
[[Category:Metric Spaces]]
3bkrpgwg5yp9kn5et5d2op3zfqsnlf6 | Uniform Convergence is Hereditary | https://proofwiki.org/wiki/Uniform_Convergence_is_Hereditary | https://proofwiki.org/wiki/Uniform_Convergence_is_Hereditary | [
"Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Sequence",
"Definition:Mapping",
"Definition:Uniform Convergence",
"Definition:Uniform Convergence",
"Definition:Metric Subspace",
"Definition:Uniform Convergence",
"Definition:Hereditary Property (Topology)",
"Definition:Metric Space"
] | [
"Category:Metric Spaces"
] |
proofwiki-11546 | Real Number to Negative Power/Integer | Let $n \in \Z$ be an integer.
Let $r^n$ be defined as $r$ to the power of $n$.
Then:
:$r^{-n} = \dfrac 1 {r^n}$ | Let $n \in \Z_{\ge 0}$.
Then from Real Number to Negative Power: Positive Integer:
:$r^{-n} = \dfrac 1 {r^n}$
It remains to show that this holds when $n < 0$.
Let $n \in \Z_{<0}$.
Then $n = - m$ for some $m \in \Z_{> 0}$.
Thus:
{{begin-eqn}}
{{eqn | l = r^{-m}
| r = \dfrac 1 {r^m}
| c = Real Number to Negat... | Let $n \in \Z$ be an [[Definition:Integer|integer]].
Let $r^n$ be defined as [[Definition:Integer Power|$r$ to the power of $n$]].
Then:
:$r^{-n} = \dfrac 1 {r^n}$ | Let $n \in \Z_{\ge 0}$.
Then from [[Real Number to Negative Power/Positive Integer|Real Number to Negative Power: Positive Integer]]:
:$r^{-n} = \dfrac 1 {r^n}$
It remains to show that this holds when $n < 0$.
Let $n \in \Z_{<0}$.
Then $n = - m$ for some $m \in \Z_{> 0}$.
Thus:
{{begin-eqn}}
{{eqn | l = r^{-m}
... | Real Number to Negative Power/Integer | https://proofwiki.org/wiki/Real_Number_to_Negative_Power/Integer | https://proofwiki.org/wiki/Real_Number_to_Negative_Power/Integer | [
"Real Number to Negative Power"
] | [
"Definition:Integer",
"Definition:Power (Algebra)/Integer"
] | [
"Real Number to Negative Power/Positive Integer",
"Real Number to Negative Power/Positive Integer",
"Definition:Reciprocal",
"Category:Real Number to Negative Power"
] |
proofwiki-11547 | Real Number to Negative Power/Positive Integer | Let $n \in \Z_{\ge 0}$ be a positive integer.
Let $r^n$ be defined as $r$ to the power of $n$.
Then:
:$r^{-n} = \dfrac 1 {r^n}$ | Proof by induction on $m$:
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$r^{-n} = \dfrac 1 {r^n}$
$\map P 0$ is the case:
{{begin-eqn}}
{{eqn | l = r^{-0}
| r = r^0
| c =
}}
{{eqn | r = 1
| c = {{Defof|Integer Power}}
}}
{{eqn | r = \dfrac 1 1
| c =
}}
{{eqn | r = \dfrac 1 {... | Let $n \in \Z_{\ge 0}$ be a [[Definition:Positive Integer|positive integer]].
Let $r^n$ be defined as [[Definition:Integer Power|$r$ to the power of $n$]].
Then:
:$r^{-n} = \dfrac 1 {r^n}$ | Proof by [[Principle of Mathematical Induction|induction]] on $m$:
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$r^{-n} = \dfrac 1 {r^n}$
$\map P 0$ is the case:
{{begin-eqn}}
{{eqn | l = r^{-0}
| r = r^0
| c =
}}
{{eqn | r = 1
| c = {{Defof|Integer P... | Real Number to Negative Power/Positive Integer | https://proofwiki.org/wiki/Real_Number_to_Negative_Power/Positive_Integer | https://proofwiki.org/wiki/Real_Number_to_Negative_Power/Positive_Integer | [
"Real Number to Negative Power"
] | [
"Definition:Positive/Integer",
"Definition:Power (Algebra)/Integer"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-11548 | Way Below iff Second Operand Preceding Supremum of Ideal implies First Operand is Element of Ideal | Let $\mathscr S = \struct {S, \preceq}$ be an up-complete ordered set.
Let $x, y \in S$.
Then $x \ll y$ {{iff}}
:$\forall I \in \map {\operatorname {Ids} } {\mathscr S}: y \preceq \sup I \implies x \in I$
where
:$\ll$ denotes the way below relation,
:$\map {\operatorname {Ids} } {\mathscr S}$ denotes the set of all ide... | === Sufficient Condition ===
Let $x \ll y$
Let $I \in \map {\operatorname {Ids} } {\mathscr S}$ such that
:$y \preceq \sup I$
By definition of ideal:
:$I$ is directed and lower.
By definition of up-complete:
:$I$ admits a supremum.
By definition of way below relation:
:$\exists i \in I: x \preceq i$
Thus by definition ... | Let $\mathscr S = \struct {S, \preceq}$ be an [[Definition:Up-Complete|up-complete]] [[Definition:Ordered Set|ordered set]].
Let $x, y \in S$.
Then $x \ll y$ {{iff}}
:$\forall I \in \map {\operatorname {Ids} } {\mathscr S}: y \preceq \sup I \implies x \in I$
where
:$\ll$ denotes the [[Definition:Element is Way Below... | === Sufficient Condition ===
Let $x \ll y$
Let $I \in \map {\operatorname {Ids} } {\mathscr S}$ such that
:$y \preceq \sup I$
By definition of [[Definition:Ideal in Ordered Set|ideal]]:
:$I$ is [[Definition:Directed Subset|directed]] and [[Definition:Lower Section|lower]].
By definition of [[Definition:Up-Complete|... | Way Below iff Second Operand Preceding Supremum of Ideal implies First Operand is Element of Ideal | https://proofwiki.org/wiki/Way_Below_iff_Second_Operand_Preceding_Supremum_of_Ideal_implies_First_Operand_is_Element_of_Ideal | https://proofwiki.org/wiki/Way_Below_iff_Second_Operand_Preceding_Supremum_of_Ideal_implies_First_Operand_is_Element_of_Ideal | [
"Way Below Relation"
] | [
"Definition:Up-Complete",
"Definition:Ordered Set",
"Definition:Element is Way Below",
"Definition:Set of Sets",
"Definition:Ideal in Ordered Set"
] | [
"Definition:Ideal in Ordered Set",
"Definition:Directed Subset",
"Definition:Lower Section",
"Definition:Up-Complete",
"Definition:Supremum of Set",
"Definition:Element is Way Below",
"Definition:Lower Section",
"Definition:Directed Subset",
"Definition:Supremum of Set",
"Definition:Supremum of Se... |
proofwiki-11549 | Lower Closure of Directed Subset is Ideal | Let $\mathscr S = \struct {S, \preceq}$ be an ordered set.
Let $D$ be a directed subset of $S$.
Then
:$D^\preceq$ is an ideal in $\mathscr S$
where $D^\preceq$ denotes the lower closure of $D$. | By Directed iff Lower Closure Directed:
:$D^\preceq$ is directed.
By Lower Closure is Lower Section:
:$D^\preceq$ is a lower section.
Thus by definition
:$D^\preceq$ is an ideal in $\mathscr S$
{{qed}} | Let $\mathscr S = \struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $D$ be a [[Definition:Directed Subset|directed subset]] of $S$.
Then
:$D^\preceq$ is an [[Definition:Ideal in Ordered Set|ideal]] in $\mathscr S$
where $D^\preceq$ denotes the [[Definition:Lower Closure of Subset|lower closure]... | By [[Directed iff Lower Closure Directed]]:
:$D^\preceq$ is [[Definition:Directed Subset|directed]].
By [[Lower Closure is Lower Section]]:
:$D^\preceq$ is a [[Definition:Lower Section|lower section]].
Thus by definition
:$D^\preceq$ is an [[Definition:Ideal in Ordered Set|ideal]] in $\mathscr S$
{{qed}} | Lower Closure of Directed Subset is Ideal | https://proofwiki.org/wiki/Lower_Closure_of_Directed_Subset_is_Ideal | https://proofwiki.org/wiki/Lower_Closure_of_Directed_Subset_is_Ideal | [
"Order Theory"
] | [
"Definition:Ordered Set",
"Definition:Directed Subset",
"Definition:Ideal in Ordered Set",
"Definition:Lower Closure/Set"
] | [
"Directed iff Lower Closure Directed",
"Definition:Directed Subset",
"Lower Closure is Lower Section",
"Definition:Lower Section",
"Definition:Ideal in Ordered Set"
] |
proofwiki-11550 | Sum from 0 to n-1 of r x^r | :$\ds \sum_{r \mathop = 0}^{n - 1} r x^r = \frac {\paren {n - 1} x^{n + 1} - n x^n + x} {\paren {x - 1}^2}$ | {{begin-eqn}}
{{eqn | l = \sum_{r \mathop = 0}^{n - 1} r x^{r - 1}
| r = \sum_{r \mathop = 0}^{n - 1} D_x x^r
| c = Power Rule for Derivatives
}}
{{eqn | r = D_x \sum_{j \mathop = 0}^{n - 1} x^r
| c = Sum Rule for Derivatives
}}
{{eqn | r = D_x \frac {x^n - 1} {x - 1}
| c = Sum of Geometric Sequ... | :$\ds \sum_{r \mathop = 0}^{n - 1} r x^r = \frac {\paren {n - 1} x^{n + 1} - n x^n + x} {\paren {x - 1}^2}$ | {{begin-eqn}}
{{eqn | l = \sum_{r \mathop = 0}^{n - 1} r x^{r - 1}
| r = \sum_{r \mathop = 0}^{n - 1} D_x x^r
| c = [[Power Rule for Derivatives]]
}}
{{eqn | r = D_x \sum_{j \mathop = 0}^{n - 1} x^r
| c = [[Sum Rule for Derivatives]]
}}
{{eqn | r = D_x \frac {x^n - 1} {x - 1}
| c = [[Sum of Geom... | Sum from 0 to n-1 of r x^r | https://proofwiki.org/wiki/Sum_from_0_to_n-1_of_r_x^r | https://proofwiki.org/wiki/Sum_from_0_to_n-1_of_r_x^r | [
"Examples of Power Series",
"Sum of Geometric Sequence"
] | [] | [
"Power Rule for Derivatives",
"Sum Rule for Derivatives",
"Sum of Geometric Sequence",
"Quotient Rule for Derivatives",
"Distributive Laws/Arithmetic",
"Category:Examples of Power Series",
"Category:Sum of Geometric Sequence"
] |
proofwiki-11551 | Defining Sequence of Natural Logarithm is Strictly Decreasing | Let $x \in \R$ be a real number such that $x > 0$.
Let $\sequence {f_n}$ be the sequence of mappings $f_n : \R_{>0} \to \R$ defined as:
:$\map {f_n} x = n \paren {\sqrt [n] x - 1}$
Then $\forall x \in \R_{>0}: \sequence {\map {f_n} x}$ is strictly decreasing. | Fix $t \in \R_{>0}$.
Then:
{{begin-eqn}}
{{eqn | l = n \paren {t^{n + 1} - 1} - \paren {n + 1} \paren {t^n - 1}
| r = \paren {t - 1}^2 \paren {1 + 2 t + 3 t^2 + \ldots + n t^{n - 1} }
| c = Sum from $0$ to $n - 1$ of $r x^r$
}}
{{eqn | o = >
| r = 0
| c = Product of Real Numbers is Positive iff ... | Let $x \in \R$ be a [[Definition:Real Number|real number]] such that $x > 0$.
Let $\sequence {f_n}$ be the [[Definition:Sequence|sequence]] of mappings $f_n : \R_{>0} \to \R$ defined as:
:$\map {f_n} x = n \paren {\sqrt [n] x - 1}$
Then $\forall x \in \R_{>0}: \sequence {\map {f_n} x}$ is [[Definition:Strictly Decre... | Fix $t \in \R_{>0}$.
Then:
{{begin-eqn}}
{{eqn | l = n \paren {t^{n + 1} - 1} - \paren {n + 1} \paren {t^n - 1}
| r = \paren {t - 1}^2 \paren {1 + 2 t + 3 t^2 + \ldots + n t^{n - 1} }
| c = [[Sum from 0 to n-1 of r x^r|Sum from $0$ to $n - 1$ of $r x^r$]]
}}
{{eqn | o = >
| r = 0
| c = [[Produc... | Defining Sequence of Natural Logarithm is Strictly Decreasing | https://proofwiki.org/wiki/Defining_Sequence_of_Natural_Logarithm_is_Strictly_Decreasing | https://proofwiki.org/wiki/Defining_Sequence_of_Natural_Logarithm_is_Strictly_Decreasing | [
"Natural Logarithms"
] | [
"Definition:Real Number",
"Definition:Sequence",
"Definition:Strictly Decreasing/Sequence"
] | [
"Sum from 0 to n-1 of r x^r",
"Product of Real Numbers is Positive iff Numbers have Same Sign",
"Power of Positive Real Number is Positive/Rational Number",
"Product of Indices of Real Number/Rational Numbers",
"Definition:Strictly Decreasing/Sequence",
"Category:Natural Logarithms"
] |
proofwiki-11552 | Logarithm Base 10 of 2 is Irrational | The common logarithm of $2$:
:$\log_{10} 2 \approx 0.30102 \, 99956 \, 63981 \, 19521 \, 37389 \ldots$
is irrational. | {{AimForCont}} $\log_{10} 2$ is rational.
Then:
{{begin-eqn}}
{{eqn | l = \log_{10} 2
| r = \frac p q
| c = for some $p, q \in \Z_{\ne 0}$
}}
{{eqn | ll= \leadsto
| l = 2
| r = 10^{p / q}
| c = {{Defof|General Logarithm}}
}}
{{eqn | ll= \leadsto
| l = 2^q
| r = 10^p
| c =... | The [[Common Logarithm of 2|common logarithm of $2$]]:
:$\log_{10} 2 \approx 0.30102 \, 99956 \, 63981 \, 19521 \, 37389 \ldots$
is [[Definition:Irrational Number|irrational]]. | {{AimForCont}} $\log_{10} 2$ is [[Definition:Rational Number|rational]].
Then:
{{begin-eqn}}
{{eqn | l = \log_{10} 2
| r = \frac p q
| c = for some $p, q \in \Z_{\ne 0}$
}}
{{eqn | ll= \leadsto
| l = 2
| r = 10^{p / q}
| c = {{Defof|General Logarithm}}
}}
{{eqn | ll= \leadsto
| l = ... | Logarithm Base 10 of 2 is Irrational/Proof 1 | https://proofwiki.org/wiki/Logarithm_Base_10_of_2_is_Irrational | https://proofwiki.org/wiki/Logarithm_Base_10_of_2_is_Irrational/Proof_1 | [
"Common Logarithms",
"2",
"10",
"Logarithm Base 10 of 2 is Irrational"
] | [
"Common Logarithm/Examples/2",
"Definition:Irrational Number"
] | [
"Definition:Rational Number",
"Definition:Power (Algebra)/Integer",
"Definition:Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Prime Factor",
"Proof by Contradiction",
"Definition:Irrational Number"
] |
proofwiki-11553 | Logarithm Base 10 of 2 is Irrational | The common logarithm of $2$:
:$\log_{10} 2 \approx 0.30102 \, 99956 \, 63981 \, 19521 \, 37389 \ldots$
is irrational. | Because $5$ is a divisor of $10$ but not $2$, it cannot be the case that $2^a = 10^b$ for $a, b \in \Z_{>0}$.
Hence this is a special case of Irrationality of Logarithm.
{{qed}} | The [[Common Logarithm of 2|common logarithm of $2$]]:
:$\log_{10} 2 \approx 0.30102 \, 99956 \, 63981 \, 19521 \, 37389 \ldots$
is [[Definition:Irrational Number|irrational]]. | Because $5$ is a [[Definition:Divisor of Integer|divisor]] of $10$ but not $2$, it cannot be the case that $2^a = 10^b$ for $a, b \in \Z_{>0}$.
Hence this is a special case of [[Irrationality of Logarithm]].
{{qed}} | Logarithm Base 10 of 2 is Irrational/Proof 2 | https://proofwiki.org/wiki/Logarithm_Base_10_of_2_is_Irrational | https://proofwiki.org/wiki/Logarithm_Base_10_of_2_is_Irrational/Proof_2 | [
"Common Logarithms",
"2",
"10",
"Logarithm Base 10 of 2 is Irrational"
] | [
"Common Logarithm/Examples/2",
"Definition:Irrational Number"
] | [
"Definition:Divisor (Algebra)/Integer",
"Irrationality of Logarithm"
] |
proofwiki-11554 | Nth Root of 1 plus x not greater than 1 plus x over n | Let $x \in \R_{>0}$ be a (strictly) positive real number.
Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Then:
:$\sqrt [n] {1 + x} \le 1 + \dfrac x n$ | From Bernoulli's Inequality:
:$\paren {1 + y}^n \ge 1 + n y$
which holds for:
:$y \in \R$ where $y > -1$
:$n \in \Z_{\ge 0}$
Thus it holds for $y \in \R_{> 0}$ and $n \in \Z_{> 0}$.
So:
{{begin-eqn}}
{{eqn | l = 1 + n y
| o = \le
| r = \paren {1 + y}^n
| c =
}}
{{eqn | ll= \leadsto
| l = 1 + n... | Let $x \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|(strictly) positive real number]].
Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Then:
:$\sqrt [n] {1 + x} \le 1 + \dfrac x n$ | From [[Bernoulli's Inequality]]:
:$\paren {1 + y}^n \ge 1 + n y$
which holds for:
:$y \in \R$ where $y > -1$
:$n \in \Z_{\ge 0}$
Thus it holds for $y \in \R_{> 0}$ and $n \in \Z_{> 0}$.
So:
{{begin-eqn}}
{{eqn | l = 1 + n y
| o = \le
| r = \paren {1 + y}^n
| c =
}}
{{eqn | ll= \leadsto
| l ... | Nth Root of 1 plus x not greater than 1 plus x over n | https://proofwiki.org/wiki/Nth_Root_of_1_plus_x_not_greater_than_1_plus_x_over_n | https://proofwiki.org/wiki/Nth_Root_of_1_plus_x_not_greater_than_1_plus_x_over_n | [
"Inequalities"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Strictly Positive/Integer"
] | [
"Bernoulli's Inequality",
"Root is Strictly Increasing"
] |
proofwiki-11555 | Reciprocal of Logarithm | Let $x, y \in \R_{> 0}$ be (strictly) positive real numbers.
Then:
:$\dfrac 1 {\log_x y} = \log_y x$ | From Primitive of $\dfrac {x^m} {\ln x}$:
:$\ds \int \frac {x^m \rd x} {\ln x} = \map \ln {\ln x} + \paren {m + 1} \ln x + \sum_{k \mathop \ge 2}^n \frac {\paren {m + 1}^k \paren {\ln x}^k} {k \times k!} + C$
The result follows by setting $m = 0$.
{{qed}} | Let $x, y \in \R_{> 0}$ be [[Definition:Strictly Positive Real Number|(strictly) positive real numbers]].
Then:
:$\dfrac 1 {\log_x y} = \log_y x$ | From [[Primitive of Power of x over Logarithm of x|Primitive of $\dfrac {x^m} {\ln x}$]]:
:$\ds \int \frac {x^m \rd x} {\ln x} = \map \ln {\ln x} + \paren {m + 1} \ln x + \sum_{k \mathop \ge 2}^n \frac {\paren {m + 1}^k \paren {\ln x}^k} {k \times k!} + C$
The result follows by setting $m = 0$.
{{qed}} | Primitive of Reciprocal of Logarithm of x/Proof 2 | https://proofwiki.org/wiki/Reciprocal_of_Logarithm | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Logarithm_of_x/Proof_2 | [
"Logarithms",
"Reciprocals",
"Reciprocal of Logarithm"
] | [
"Definition:Strictly Positive/Real Number"
] | [
"Primitive of Power of x over Logarithm of x"
] |
proofwiki-11556 | Reciprocal of Logarithm | Let $x, y \in \R_{> 0}$ be (strictly) positive real numbers.
Then:
:$\dfrac 1 {\log_x y} = \log_y x$ | {{begin-eqn}}
{{eqn | l = \log_x y \log_y x
| r = \log_y y
| c = Change of Base of Logarithm
}}
{{eqn | r = 1
| c =
}}
{{eqn | ll= \leadsto
| l = \log_y x
| r = \dfrac 1 {\log_x y}
| c =
}}
{{end-eqn}}
{{qed}} | Let $x, y \in \R_{> 0}$ be [[Definition:Strictly Positive Real Number|(strictly) positive real numbers]].
Then:
:$\dfrac 1 {\log_x y} = \log_y x$ | {{begin-eqn}}
{{eqn | l = \log_x y \log_y x
| r = \log_y y
| c = [[Change of Base of Logarithm]]
}}
{{eqn | r = 1
| c =
}}
{{eqn | ll= \leadsto
| l = \log_y x
| r = \dfrac 1 {\log_x y}
| c =
}}
{{end-eqn}}
{{qed}} | Reciprocal of Logarithm | https://proofwiki.org/wiki/Reciprocal_of_Logarithm | https://proofwiki.org/wiki/Reciprocal_of_Logarithm | [
"Logarithms",
"Reciprocals",
"Reciprocal of Logarithm"
] | [
"Definition:Strictly Positive/Real Number"
] | [
"Change of Base of Logarithm"
] |
proofwiki-11557 | Discontinuity of Monotonic Function is Jump Discontinuity | Let $X$ be an open set of $\R$.
Let $f: X \to Y$ be a monotone real function.
Then $f$ is discontinuous at a point $c \in X$ {{iff}} $c$ is a jump discontinuity of $f$. | The backwards implication follows directly from definition of a jump discontinuity.
{{qed|lemma}}
For the forwards implication, we prove the contrapositive.
{{WLOG}} suppose $f$ is increasing.
If $f$ is decreasing, note that $-f$ is increasing and we can simply replace $f$ by $-f$ in the following analysis.
Suppose tha... | Let $X$ be an [[Definition:Open Set (Real Analysis)|open set]] of $\R$.
Let $f: X \to Y$ be a [[Definition:Monotone Real Function|monotone real function]].
Then $f$ is [[Definition:Discontinuous|discontinuous]] at a point $c \in X$ {{iff}} $c$ is a [[Definition:Jump Discontinuity|jump discontinuity]] of $f$. | The backwards implication follows directly from definition of a [[Definition:Jump Discontinuity|jump discontinuity]].
{{qed|lemma}}
For the forwards implication, we prove the [[Definition:Contrapositive Statement|contrapositive]].
{{WLOG}} suppose $f$ is [[Definition:Increasing Real Function|increasing]].
If $f$ is... | Discontinuity of Monotonic Function is Jump Discontinuity | https://proofwiki.org/wiki/Discontinuity_of_Monotonic_Function_is_Jump_Discontinuity | https://proofwiki.org/wiki/Discontinuity_of_Monotonic_Function_is_Jump_Discontinuity | [
"Monotone Real Functions",
"Real Analysis",
"Proofs by Contraposition"
] | [
"Definition:Open Set/Real Analysis",
"Definition:Monotone (Order Theory)/Real Function",
"Definition:Discontinuous",
"Definition:Discontinuity (Real Analysis)/Jump"
] | [
"Definition:Discontinuity (Real Analysis)/Jump",
"Definition:Contrapositive Statement",
"Definition:Increasing/Real Function",
"Definition:Decreasing/Real Function",
"Definition:Increasing/Real Function",
"Definition:Discontinuity (Real Analysis)/Jump",
"Definition:Discontinuity (Real Analysis)/Jump",
... |
proofwiki-11558 | Surjective Monotone Function is Continuous | Let $X$ be an open set of $\R$.
Let $Y$ be a real interval.
Let $f: X \to Y$ be a surjective monotone real function.
Then $f$ is continuous on $X$. | {{Tidy}}
{{MissingLinks}}
{{WLOG}}, let $f$ be increasing.
Let $c \in X$.
From Limit of Monotone Real Function: Corollary, the one sided limits of monotone functions exist:
{{begin-eqn}}
{{eqn | l = L^-_c
| m = \lim_{x \mathop \to c^-} \map f x
| mo= =
| r = \sup_{x \mathop < c} \map f x
}}
{{eqn | l ... | Let $X$ be an [[Definition:Open Set (Real Analysis)|open set]] of $\R$.
Let $Y$ be a [[Definition:Real Interval|real interval]].
Let $f: X \to Y$ be a [[Definition:Surjection|surjective]] [[Definition:Monotone Real Function|monotone real function]].
Then $f$ is [[Definition:Continuous on Interval|continuous]] on $X... | {{Tidy}}
{{MissingLinks}}
{{WLOG}}, let $f$ be [[Definition:Increasing Real Function|increasing]].
Let $c \in X$.
From [[Limit of Monotone Real Function/Increasing/Corollary|Limit of Monotone Real Function: Corollary]], the one sided limits of monotone functions exist:
{{begin-eqn}}
{{eqn | l = L^-_c
| m = \l... | Surjective Monotone Function is Continuous | https://proofwiki.org/wiki/Surjective_Monotone_Function_is_Continuous | https://proofwiki.org/wiki/Surjective_Monotone_Function_is_Continuous | [
"Monotone Real Functions",
"Real Analysis",
"Continuity"
] | [
"Definition:Open Set/Real Analysis",
"Definition:Real Interval",
"Definition:Surjection",
"Definition:Monotone (Order Theory)/Real Function",
"Definition:Continuous Real Function/Interval"
] | [
"Definition:Increasing/Real Function",
"Limit of Increasing Function/Corollary",
"Limit iff Limits from Left and Right",
"Definition:Increasing/Real Function",
"Discontinuity of Monotonic Function is Jump Discontinuity",
"Definition:Discontinuity (Real Analysis)/Jump",
"Real Numbers are Densely Ordered"... |
proofwiki-11559 | Upper Bound of Natural Logarithm/Corollary | :$\forall s \in \R_{>0}: \ln x \le \dfrac {x^s} s$ | {{begin-eqn}}
{{eqn | l = s \ln x
| r = \ln {x^s}
| c = Logarithm of Power
}}
{{eqn | o = \le
| r = x^s - 1
| c = Upper Bound of Natural Logarithm
}}
{{eqn | o = \le
| r = x^s
| c =
}}
{{end-eqn}}
The result follows by dividing both sides by $s$.
{{qed}} | :$\forall s \in \R_{>0}: \ln x \le \dfrac {x^s} s$ | {{begin-eqn}}
{{eqn | l = s \ln x
| r = \ln {x^s}
| c = [[Logarithm of Power]]
}}
{{eqn | o = \le
| r = x^s - 1
| c = [[Upper Bound of Natural Logarithm]]
}}
{{eqn | o = \le
| r = x^s
| c =
}}
{{end-eqn}}
The result follows by dividing both sides by $s$.
{{qed}} | Upper Bound of Natural Logarithm/Corollary | https://proofwiki.org/wiki/Upper_Bound_of_Natural_Logarithm/Corollary | https://proofwiki.org/wiki/Upper_Bound_of_Natural_Logarithm/Corollary | [
"Upper Bound of Natural Logarithm"
] | [] | [
"Logarithm of Power",
"Upper Bound of Natural Logarithm"
] |
proofwiki-11560 | Powers Drown Logarithms/Corollary | :$\ds \lim_{y \mathop \to 0_+} y^r \ln y = 0$ | Put $y = \dfrac 1 x$ in the Powers Drown Logarithms.
{{qed}} | :$\ds \lim_{y \mathop \to 0_+} y^r \ln y = 0$ | Put $y = \dfrac 1 x$ in the [[Powers Drown Logarithms]].
{{qed}} | Powers Drown Logarithms/Corollary | https://proofwiki.org/wiki/Powers_Drown_Logarithms/Corollary | https://proofwiki.org/wiki/Powers_Drown_Logarithms/Corollary | [
"Logarithms",
"Powers"
] | [] | [
"Powers Drown Logarithms"
] |
proofwiki-11561 | Logarithm of Logarithm in terms of Natural Logarithms | Let $b, x \in \R_{>0}$ be (strictly) positive real numbers.
Then:
:$\map {\log_b} {\log_b x} = \dfrac {\map \ln {\ln x} - \map \ln {\ln b} } {\ln b}$
where $\ln x$ denotes the natural logarithm of $x$. | {{begin-eqn}}
{{eqn | l = \map {\log_b} {\log_b x}
| r = \map {\log_b} {\frac {\ln x} {\ln b} }
| c = Change of Base of Logarithm
}}
{{eqn | r = \map {\log_b} {\ln x} - \map {\log_b} {\ln b}
| c = Difference of Logarithms
}}
{{eqn | r = \frac {\map \ln {\ln x} } {\ln b} - \frac {\map \ln {\ln b} } {\l... | Let $b, x \in \R_{>0}$ be [[Definition:Strictly Positive Real Number|(strictly) positive real numbers]].
Then:
:$\map {\log_b} {\log_b x} = \dfrac {\map \ln {\ln x} - \map \ln {\ln b} } {\ln b}$
where $\ln x$ denotes the [[Definition:Natural Logarithm|natural logarithm]] of $x$. | {{begin-eqn}}
{{eqn | l = \map {\log_b} {\log_b x}
| r = \map {\log_b} {\frac {\ln x} {\ln b} }
| c = [[Change of Base of Logarithm]]
}}
{{eqn | r = \map {\log_b} {\ln x} - \map {\log_b} {\ln b}
| c = [[Difference of Logarithms]]
}}
{{eqn | r = \frac {\map \ln {\ln x} } {\ln b} - \frac {\map \ln {\ln ... | Logarithm of Logarithm in terms of Natural Logarithms | https://proofwiki.org/wiki/Logarithm_of_Logarithm_in_terms_of_Natural_Logarithms | https://proofwiki.org/wiki/Logarithm_of_Logarithm_in_terms_of_Natural_Logarithms | [
"Logarithms",
"Natural Logarithms"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Natural Logarithm"
] | [
"Change of Base of Logarithm",
"Difference of Logarithms",
"Change of Base of Logarithm"
] |
proofwiki-11562 | Limit of Bounded Convergent Sequence is Bounded | Let $\sequence {x_n}$, $\sequence {a_n}$, and $\sequence {b_n}$ be convergent sequences in $\R$.
Let $\sequence {x_n}$, $\sequence {a_n}$, and $\sequence {b_n}$ converge to $x, a, b \in \R$, respectively.
Suppose that:
:$\exists N \in \N: n \ge N \implies a_n \le x_n \le b_n$
Then:
:$a \le x \le b$ | {{AimForCont}} that $x < a$.
Let $\epsilon = \dfrac {a - x} 2 > 0$
From the convergence of $\sequence {x_n}$:
:$\exists M_1 \in \N : n \ge M \implies x - \epsilon < x_n < x + \epsilon$
Or, equivalently:
:$\exists M_1 \in \N : n \ge M \implies \dfrac {3 x - a} 2 < x_n < \dfrac {x + a} 2$
From the convergence of $\sequen... | Let $\sequence {x_n}$, $\sequence {a_n}$, and $\sequence {b_n}$ be [[Definition:Convergent Real Sequence|convergent sequences in $\R$]].
Let $\sequence {x_n}$, $\sequence {a_n}$, and $\sequence {b_n}$ [[Definition:Convergent Real Sequence|converge]] to $x, a, b \in \R$, respectively.
Suppose that:
:$\exists N \in \N:... | {{AimForCont}} that $x < a$.
Let $\epsilon = \dfrac {a - x} 2 > 0$
From the convergence of $\sequence {x_n}$:
:$\exists M_1 \in \N : n \ge M \implies x - \epsilon < x_n < x + \epsilon$
Or, [[Definition:Logical Equivalence|equivalently]]:
:$\exists M_1 \in \N : n \ge M \implies \dfrac {3 x - a} 2 < x_n < \dfrac {x + ... | Limit of Bounded Convergent Sequence is Bounded | https://proofwiki.org/wiki/Limit_of_Bounded_Convergent_Sequence_is_Bounded | https://proofwiki.org/wiki/Limit_of_Bounded_Convergent_Sequence_is_Bounded | [
"Limits of Sequences",
"Real Analysis"
] | [
"Definition:Convergent Sequence/Real Numbers",
"Definition:Convergent Sequence/Real Numbers"
] | [
"Definition:Logical Equivalence",
"Definition:Logical Equivalence",
"Definition:Contradiction",
"Definition:Mutatis Mutandis",
"Definition:Contradiction",
"Proof by Contradiction",
"Category:Limits of Sequences",
"Category:Real Analysis"
] |
proofwiki-11563 | Logarithm of Power/Natural Logarithm/Natural Power | Let $x \in \R$ be a strictly positive real number.
Let $n \in \Z_{\ge 0}$ be any natural number.
Let $\ln x$ be the natural logarithm of $x$.
Then:
:$\map \ln {x^n} = n \ln x$ | Proof by Mathematical Induction:
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$\forall x \in \R_{>0}: \map \ln {x^n} = n \ln x$ | Let $x \in \R$ be a [[Definition:Strictly Positive Real Number|strictly positive real number]].
Let $n \in \Z_{\ge 0}$ be any [[Definition:Natural Numbers|natural number]].
Let $\ln x$ be the [[Definition:Natural Logarithm|natural logarithm]] of $x$.
Then:
:$\map \ln {x^n} = n \ln x$ | Proof by [[Proof by Mathematical Induction|Mathematical Induction]]:
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\forall x \in \R_{>0}: \map \ln {x^n} = n \ln x$ | Logarithm of Power/Natural Logarithm/Natural Power | https://proofwiki.org/wiki/Logarithm_of_Power/Natural_Logarithm/Natural_Power | https://proofwiki.org/wiki/Logarithm_of_Power/Natural_Logarithm/Natural_Power | [
"Logarithm of Power"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Natural Numbers",
"Definition:Natural Logarithm"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition"
] |
proofwiki-11564 | Logarithm of Power/Natural Logarithm/Integer Power | Let $x \in \R$ be a strictly positive real number.
Let $n \in \R$ be any integer.
Let $\ln x$ be the natural logarithm of $x$.
Then:
:$\map \ln {x^n} = n \ln x$ | From Logarithm of Power/Natural Logarithm/Natural Power, the theorem is already proven for positive integers.
Let $j \in \Z_{<0}$.
Let $-j = k \in Z_{>0}$.
Then:
{{begin-eqn}}
{{eqn | l = 0
| r = \ln 1
| c = Logarithm of 1 is 0
}}
{{eqn | r = \map \ln {x^k x^{-k} }
}}
{{eqn | r = \map \ln {x^k} + \map \ln {... | Let $x \in \R$ be a [[Definition:Strictly Positive Real Number|strictly positive real number]].
Let $n \in \R$ be any [[Definition:Integer|integer]].
Let $\ln x$ be the [[Definition:Natural Logarithm|natural logarithm]] of $x$.
Then:
:$\map \ln {x^n} = n \ln x$ | From [[Logarithm of Power/Natural Logarithm/Natural Power]], the theorem is already proven for positive integers.
Let $j \in \Z_{<0}$.
Let $-j = k \in Z_{>0}$.
Then:
{{begin-eqn}}
{{eqn | l = 0
| r = \ln 1
| c = [[Logarithm of 1 is 0]]
}}
{{eqn | r = \map \ln {x^k x^{-k} }
}}
{{eqn | r = \map \ln {x^k} +... | Logarithm of Power/Natural Logarithm/Integer Power | https://proofwiki.org/wiki/Logarithm_of_Power/Natural_Logarithm/Integer_Power | https://proofwiki.org/wiki/Logarithm_of_Power/Natural_Logarithm/Integer_Power | [
"Logarithm of Power"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Integer",
"Definition:Natural Logarithm"
] | [
"Logarithm of Power/Natural Logarithm/Natural Power",
"Natural Logarithm of 1 is 0",
"Logarithm of Power/Natural Logarithm/Natural Power",
"Category:Logarithm of Power"
] |
proofwiki-11565 | Logarithm of Power/Natural Logarithm/Rational Power | Let $x \in \R$ be a strictly positive real number.
Let $r \in \R$ be any rational number.
Let $\ln x$ be the natural logarithm of $x$.
Then:
:$\map \ln {x^r} = r \ln x$ | Let $r = \dfrac s t$, where $s \in \Z$ and $t \in \Z_{>0}$.
First:
{{begin-eqn}}
{{eqn | l = \map \ln x
| r = \map \ln {x^{t / t} }
}}
{{eqn | r = \map \ln {\paren {x^{1 / t} }^t}
| c = Product of Indices of Real Number/Rational Numbers
}}
{{eqn | r = t \map \ln {x^{1 / t} }
| c = Logarithm of Power/N... | Let $x \in \R$ be a [[Definition:Strictly Positive Real Number|strictly positive real number]].
Let $r \in \R$ be any [[Definition:Rational Number|rational number]].
Let $\ln x$ be the [[Definition:Natural Logarithm|natural logarithm]] of $x$.
Then:
:$\map \ln {x^r} = r \ln x$ | Let $r = \dfrac s t$, where $s \in \Z$ and $t \in \Z_{>0}$.
First:
{{begin-eqn}}
{{eqn | l = \map \ln x
| r = \map \ln {x^{t / t} }
}}
{{eqn | r = \map \ln {\paren {x^{1 / t} }^t}
| c = [[Product of Indices of Real Number/Rational Numbers]]
}}
{{eqn | r = t \map \ln {x^{1 / t} }
| c = [[Logarithm of ... | Logarithm of Power/Natural Logarithm/Rational Power | https://proofwiki.org/wiki/Logarithm_of_Power/Natural_Logarithm/Rational_Power | https://proofwiki.org/wiki/Logarithm_of_Power/Natural_Logarithm/Rational_Power | [
"Logarithm of Power"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Rational Number",
"Definition:Natural Logarithm"
] | [
"Product of Indices of Real Number/Rational Numbers",
"Logarithm of Power/Natural Logarithm/Integer Power",
"Product of Indices of Real Number/Rational Numbers",
"Logarithm of Power/Natural Logarithm/Integer Power",
"Category:Logarithm of Power"
] |
proofwiki-11566 | Dini's Theorem | Let $K \subseteq \R$ be compact.
Let $\sequence {f_n}$ be a sequence of continuous real functions defined on $K$.
Let $\sequence {f_n}$ converge pointwise to a continuous function $f$.
Suppose that:
:$\forall x \in K : \sequence {\map {f_n} x}$ is monotone.
Then the convergence of $\sequence {f_n}$ to $f$ is uniform. | For each $n$, define $d_n : K \to \R$ by:
:$d_n := \size {f_n - f}$
We have:
:$(1): \quad \sequence {d_n}$ converge pointwise to $0$
since $\sequence {f_n}$ converges pointwise to $f$.
We also have:
:$(2): \quad \forall x \in K : \sequence {\map {d_n} x}$ is monotonically decreasing
since $\sequence {\map {f_n} x}$ is ... | Let $K \subseteq \R$ be [[Definition:Compact Metric Space|compact]].
Let $\sequence {f_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Continuous Real Function|continuous real functions]] defined on $K$.
Let $\sequence {f_n}$ [[Definition:Pointwise Convergence|converge pointwise]] to a [[Definition:Continuo... | For each $n$, define $d_n : K \to \R$ by:
:$d_n := \size {f_n - f}$
We have:
:$(1): \quad \sequence {d_n}$ [[Definition:Pointwise Convergence|converge pointwise]] to $0$
since $\sequence {f_n}$ [[Definition:Pointwise Convergence|converges pointwise]] to $f$.
We also have:
:$(2): \quad \forall x \in K : \sequence {\ma... | Dini's Theorem | https://proofwiki.org/wiki/Dini's_Theorem | https://proofwiki.org/wiki/Dini's_Theorem | [
"Real Analysis"
] | [
"Definition:Compact Space/Metric Space",
"Definition:Sequence",
"Definition:Continuous Real Function",
"Definition:Pointwise Convergence",
"Definition:Continuous Real Function",
"Definition:Monotone (Order Theory)/Sequence",
"Definition:Uniform Convergence"
] | [
"Definition:Pointwise Convergence",
"Definition:Pointwise Convergence",
"Definition:Decreasing/Sequence/Real Sequence",
"Definition:Monotone (Order Theory)/Sequence",
"Definition:Continuous Real Function",
"Definition:Open Set",
"Definition:Compact Space/Metric Space",
"Definition:Subcover/Finite",
... |
proofwiki-11567 | Way Below in Meet-Continuous Lattice | Let $\mathscr S = \struct {S, \vee, \wedge, \preceq}$ be a meet-continuous bounded below lattice.
Let $x, y \in S$.
Then $x \ll y$ {{iff}}
:$\forall I \in \map {\operatorname {Ids} } {\mathscr S}: y = \sup I \implies x \in I$
where
:$\ll$ denotes the way below relation,
:$\map {\operatorname {Ids} } {\mathscr S}$ denot... | === Sufficient Condition ===
Let $x \ll y$
Let $I \in \map {\operatorname {Ids} } {\mathscr S}$ such that
:$y = \sup I$
By definition of reflexivity:
:$y \preceq \sup I$
By definition of meet-continuous:
:$\mathscr S$ is up-complete.
Thus by Way Below iff Second Operand Preceding Supremum of Ideal implies First Operand... | Let $\mathscr S = \struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Meet-Continuous Lattice|meet-continuous]] [[Definition:Bounded Below|bounded below]] [[Definition:Lattice (Order Theory)|lattice]].
Let $x, y \in S$.
Then $x \ll y$ {{iff}}
:$\forall I \in \map {\operatorname {Ids} } {\mathscr S}: y = \sup I \im... | === Sufficient Condition ===
Let $x \ll y$
Let $I \in \map {\operatorname {Ids} } {\mathscr S}$ such that
:$y = \sup I$
By definition of [[Definition:Reflexivity|reflexivity]]:
:$y \preceq \sup I$
By definition of [[Definition:Meet-Continuous Lattice|meet-continuous]]:
:$\mathscr S$ is [[Definition:Up-Complete|up-c... | Way Below in Meet-Continuous Lattice | https://proofwiki.org/wiki/Way_Below_in_Meet-Continuous_Lattice | https://proofwiki.org/wiki/Way_Below_in_Meet-Continuous_Lattice | [
"Way Below Relation",
"Meet-Continuous Lattices"
] | [
"Definition:Meet-Continuous Lattice",
"Definition:Bounded Below",
"Definition:Lattice (Order Theory)",
"Definition:Element is Way Below",
"Definition:Set of Sets",
"Definition:Ideal in Ordered Set"
] | [
"Definition:Reflexivity",
"Definition:Meet-Continuous Lattice",
"Definition:Up-Complete",
"Way Below iff Second Operand Preceding Supremum of Ideal implies First Operand is Element of Ideal",
"Definition:Up-Complete",
"Way Below iff Second Operand Preceding Supremum of Ideal implies First Operand is Eleme... |
proofwiki-11568 | Approximation to Binary Logarithm from Natural and Common Logarithm | The binary logarithm $\lg x$ can be approximated, to within $1 \%$, by the expression:
:$\lg x \approx \ln x + \log_{10} x$
That is, by the sum of the natural logarithm and common logarithm. | {{begin-eqn}}
{{eqn | l = \lg x
| r = \frac {\ln x} {\ln 2}
| c = Change of Base of Logarithm
}}
{{eqn | r = \frac {\log_{10} x} {\log_{10} 2}
| c = Change of Base of Logarithm
}}
{{eqn | ll= \leadsto
| l = \lg x
| r = \frac {\ln x + \log_{10} x} {\ln 2 + \log_{10} 2}
| c =
}}
{{eq... | The [[Definition:Binary Logarithm|binary logarithm]] $\lg x$ can be approximated, to within $1 \%$, by the expression:
:$\lg x \approx \ln x + \log_{10} x$
That is, by the sum of the [[Definition:Natural Logarithm|natural logarithm]] and [[Definition:Common Logarithm|common logarithm]]. | {{begin-eqn}}
{{eqn | l = \lg x
| r = \frac {\ln x} {\ln 2}
| c = [[Change of Base of Logarithm]]
}}
{{eqn | r = \frac {\log_{10} x} {\log_{10} 2}
| c = [[Change of Base of Logarithm]]
}}
{{eqn | ll= \leadsto
| l = \lg x
| r = \frac {\ln x + \log_{10} x} {\ln 2 + \log_{10} 2}
| c = ... | Approximation to Binary Logarithm from Natural and Common Logarithm | https://proofwiki.org/wiki/Approximation_to_Binary_Logarithm_from_Natural_and_Common_Logarithm | https://proofwiki.org/wiki/Approximation_to_Binary_Logarithm_from_Natural_and_Common_Logarithm | [
"Logarithms"
] | [
"Definition:General Logarithm/Binary",
"Definition:Natural Logarithm",
"Definition:General Logarithm/Common"
] | [
"Change of Base of Logarithm",
"Change of Base of Logarithm",
"Change of Base of Logarithm",
"Reciprocal of Logarithm"
] |
proofwiki-11569 | Value of b for b by Logarithm Base b of x to be Minimum | Let $x \in \R_{> 0}$ be a (strictly) positive real number.
Consider the real function $f: \R_{> 0} \to \R$ defined as:
:$\map f b := b \log_b x$
$f$ attains a minimum when
:$b = e$
where $e$ is Euler's number. | From the Interior Extremum Theorem, when $f$ is at a minimum, its derivative $\dfrac \d {\d b} f$ will be zero.
Let $y = \map f b$.
We have:
{{begin-eqn}}
{{eqn | l = y
| r = b \log_b x
| c =
}}
{{eqn | r = \frac {b \ln x} {\ln b}
| c = Change of Base of Logarithm
}}
{{eqn | ll= \leadsto
| l =... | Let $x \in \R_{> 0}$ be a [[Definition:Strictly Positive Real Number|(strictly) positive real number]].
Consider the [[Definition:Real Function|real function]] $f: \R_{> 0} \to \R$ defined as:
:$\map f b := b \log_b x$
$f$ attains a [[Definition:Local Minimum|minimum]] when
:$b = e$
where $e$ is [[Definition:Euler's... | From the [[Interior Extremum Theorem]], when $f$ is at a [[Definition:Local Minimum|minimum]], its [[Definition:Derivative|derivative]] $\dfrac \d {\d b} f$ will be zero.
Let $y = \map f b$.
We have:
{{begin-eqn}}
{{eqn | l = y
| r = b \log_b x
| c =
}}
{{eqn | r = \frac {b \ln x} {\ln b}
| c = ... | Value of b for b by Logarithm Base b of x to be Minimum | https://proofwiki.org/wiki/Value_of_b_for_b_by_Logarithm_Base_b_of_x_to_be_Minimum | https://proofwiki.org/wiki/Value_of_b_for_b_by_Logarithm_Base_b_of_x_to_be_Minimum | [
"Logarithms"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Real Function",
"Definition:Minimum Value of Real Function/Local",
"Definition:Euler's Number"
] | [
"Interior Extremum Theorem",
"Definition:Minimum Value of Real Function/Local",
"Definition:Derivative",
"Change of Base of Logarithm",
"Quotient Rule for Derivatives",
"Derivative of Natural Logarithm Function",
"Derivative of Identity Function",
"Definition:Minimum Value of Real Function/Local",
"... |
proofwiki-11570 | Monotone Real Function with Everywhere Dense Image is Continuous | Let $I$ and $J$ be real intervals.
Let $f: I \to J$ be a monotone real function.
Let $f \sqbrk I$ be everywhere dense in $J$, where $f \sqbrk I$ denotes the image of $I$ under $f$.
Then $f$ is continuous on $I$. | {{WLOG}}, let $f$ be increasing.
{{AimForCont}} $f$ is discontinuous at $c \in I$.
Let:
:$L^-$ denote $\ds \lim_{x \mathop \to c^-} \map f x$
:$L$ denote $\map f c$
:$L^+$ denote $\ds \lim_{x \mathop \to c^+} \map f x$
From Discontinuity of Monotonic Function is Jump Discontinuity, $L^-$ and $L^+$ are (finite) real num... | Let $I$ and $J$ be [[Definition:Real Interval|real intervals]].
Let $f: I \to J$ be a [[Definition:Monotone Real Function|monotone real function]].
Let $f \sqbrk I$ be [[Definition:Everywhere Dense|everywhere dense]] in $J$, where $f \sqbrk I$ denotes the [[Definition:Image of Mapping|image]] of $I$ under $f$.
Then... | {{WLOG}}, let $f$ be [[Definition:Increasing Real Function|increasing]].
{{AimForCont}} $f$ is [[Definition:Discontinuous|discontinuous]] at $c \in I$.
Let:
:$L^-$ denote $\ds \lim_{x \mathop \to c^-} \map f x$
:$L$ denote $\map f c$
:$L^+$ denote $\ds \lim_{x \mathop \to c^+} \map f x$
From [[Discontinuity of Monot... | Monotone Real Function with Everywhere Dense Image is Continuous | https://proofwiki.org/wiki/Monotone_Real_Function_with_Everywhere_Dense_Image_is_Continuous | https://proofwiki.org/wiki/Monotone_Real_Function_with_Everywhere_Dense_Image_is_Continuous | [
"Continuous Real Functions",
"Monotone Real Functions",
"Real Analysis"
] | [
"Definition:Real Interval",
"Definition:Monotone (Order Theory)/Real Function",
"Definition:Everywhere Dense",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Continuous Real Function"
] | [
"Definition:Increasing/Real Function",
"Definition:Discontinuous",
"Discontinuity of Monotonic Function is Jump Discontinuity",
"Definition:Real Number",
"Definition:Discontinuous",
"Definition:Contradiction",
"Definition:Degenerate Connected Set/Non-Degenerate",
"Non-Degenerate Real Interval is Infin... |
proofwiki-11571 | Relative Difference between Infinite Set and Finite Set is Infinite | Let $S$ be an infinite set.
Let $T$ be a finite set.
Then $S \setminus T$ is an infinite set. | {{AimForCont}} $S \setminus T$ is a finite set.
Then:
{{begin-eqn}}
{{eqn | l = S
| o = \subseteq
| r = S \cup T
| c = Set is Subset of Union
}}
{{eqn | r = \paren {S \setminus T} \cup T
| c = Set Difference Union Second Set is Union
}}
{{end-eqn}}
By Union of Finite Sets is Finite, $S \cup T$ i... | Let $S$ be an [[Definition:Infinite Set|infinite set]].
Let $T$ be a [[Definition:Finite Set|finite set]].
Then $S \setminus T$ is an [[Definition:Infinite Set|infinite set]]. | {{AimForCont}} $S \setminus T$ is a [[Definition:Finite Set|finite set]].
Then:
{{begin-eqn}}
{{eqn | l = S
| o = \subseteq
| r = S \cup T
| c = [[Set is Subset of Union]]
}}
{{eqn | r = \paren {S \setminus T} \cup T
| c = [[Set Difference Union Second Set is Union]]
}}
{{end-eqn}}
By [[Union ... | Relative Difference between Infinite Set and Finite Set is Infinite | https://proofwiki.org/wiki/Relative_Difference_between_Infinite_Set_and_Finite_Set_is_Infinite | https://proofwiki.org/wiki/Relative_Difference_between_Infinite_Set_and_Finite_Set_is_Infinite | [
"Set Difference",
"Infinite Sets"
] | [
"Definition:Infinite Set",
"Definition:Finite Set",
"Definition:Infinite Set"
] | [
"Definition:Finite Set",
"Set is Subset of Union",
"Set Difference Union Second Set is Union",
"Union of Finite Sets is Finite",
"Definition:Finite Set",
"Subset of Finite Set is Finite",
"Definition:Finite Set",
"Definition:Infinite Set",
"Definition:Contradiction",
"Definition:Infinite Set",
"... |
proofwiki-11572 | Set Difference over Subset | Let $A$, $B$, and $S$ be sets.
Let $A \subseteq B$.
Then:
:$A \setminus S \subseteq B \setminus S$ | {{begin-eqn}}
{{eqn | l = A \setminus S
| r = A \cap \map \complement S
| c = Set Difference as Intersection with Complement
}}
{{eqn | o = \subseteq
| r = B \cap \map \complement S
| c = {{Corollary|Set Intersection Preserves Subsets}}
}}
{{eqn | r = B \setminus S
| c = Set Difference as ... | Let $A$, $B$, and $S$ be [[Definition:Set|sets]].
Let $A \subseteq B$.
Then:
:$A \setminus S \subseteq B \setminus S$ | {{begin-eqn}}
{{eqn | l = A \setminus S
| r = A \cap \map \complement S
| c = [[Set Difference as Intersection with Complement]]
}}
{{eqn | o = \subseteq
| r = B \cap \map \complement S
| c = {{Corollary|Set Intersection Preserves Subsets}}
}}
{{eqn | r = B \setminus S
| c = [[Set Differen... | Set Difference over Subset | https://proofwiki.org/wiki/Set_Difference_over_Subset | https://proofwiki.org/wiki/Set_Difference_over_Subset | [
"Set Difference"
] | [
"Definition:Set"
] | [
"Set Difference as Intersection with Complement",
"Set Difference as Intersection with Complement",
"Category:Set Difference"
] |
proofwiki-11573 | Subset of Empty Set iff Empty | Let $S$ be a set.
Let $\O$ denote the empty set.
Then $S \subseteq \O$ {{iff}} $S = \O$. | Suppose $x \in S$.
Then since $S \subseteq \O$, it follows that $x \in \O$.
Hence $x \notin S$.
That is, $S = \O$.
{{qed}}
Category:Empty Set
h7lpbqjhov78wp3cpmrvdho7zc9riec | Let $S$ be a [[Definition:Set|set]].
Let $\O$ denote the [[Definition:Empty Set|empty set]].
Then $S \subseteq \O$ {{iff}} $S = \O$. | Suppose $x \in S$.
Then since $S \subseteq \O$, it follows that $x \in \O$.
Hence $x \notin S$.
That is, $S = \O$.
{{qed}}
[[Category:Empty Set]]
h7lpbqjhov78wp3cpmrvdho7zc9riec | Subset of Empty Set iff Empty | https://proofwiki.org/wiki/Subset_of_Empty_Set_iff_Empty | https://proofwiki.org/wiki/Subset_of_Empty_Set_iff_Empty | [
"Empty Set"
] | [
"Definition:Set",
"Definition:Empty Set"
] | [
"Category:Empty Set"
] |
proofwiki-11574 | Finite Subset of Metric Space is Closed | Let $M = \struct {A, d}$ be a metric space.
Let $S \subseteq A$ be finite.
Then $S$ is closed in $M$. | From Metric Space is Hausdorff, $M$ is Hausdorff.
From Finite Subspace of Hausdorff Space is Closed, $S$ is closed.
{{qed}}
Category:Closed Sets
Category:Metric Spaces
o2ifexi1cunsg6pq68v9xoyi184umf8 | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $S \subseteq A$ be [[Definition:Finite Set|finite]].
Then $S$ is [[Definition:Closed Set (Metric Space)|closed]] in $M$. | From [[Metric Space is Hausdorff]], $M$ is [[Definition:Hausdorff Space|Hausdorff]].
From [[Finite Subspace of Hausdorff Space is Closed]], $S$ is [[Definition:Closed Set (Metric Space)|closed]].
{{qed}}
[[Category:Closed Sets]]
[[Category:Metric Spaces]]
o2ifexi1cunsg6pq68v9xoyi184umf8 | Finite Subset of Metric Space is Closed | https://proofwiki.org/wiki/Finite_Subset_of_Metric_Space_is_Closed | https://proofwiki.org/wiki/Finite_Subset_of_Metric_Space_is_Closed | [
"Closed Sets",
"Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Finite Set",
"Definition:Closed Set/Metric Space"
] | [
"Metric Space is T2",
"Definition:T2 Space",
"Compact Subspace of Hausdorff Space is Closed/Corollary",
"Definition:Closed Set/Metric Space",
"Category:Closed Sets",
"Category:Metric Spaces"
] |
proofwiki-11575 | Monotone Real Function with Everywhere Dense Image is Continuous/Lemma | :$\ds \openint {\lim_{x \mathop \to c^-} \map f x} {\lim_{x \mathop \to c^+} \map f x} \cap f \sqbrk I \subseteq \set {\map f c}$ | From Discontinuity of Monotonic Function is Jump Discontinuity, $\ds \lim_{x \mathop \to c^-} \map f x$ and $\ds \lim_{x \mathop \to c^+} \map f x$ are finite.
Since $f$ is increasing:
:$\ds \lim_{x \mathop \to c^-} \map f x < \lim_{x \mathop \to c^+} \map f x$
Suppose $z \in \ds \openint {\lim_{x \mathop \to c^-} \map... | :$\ds \openint {\lim_{x \mathop \to c^-} \map f x} {\lim_{x \mathop \to c^+} \map f x} \cap f \sqbrk I \subseteq \set {\map f c}$ | From [[Discontinuity of Monotonic Function is Jump Discontinuity]], $\ds \lim_{x \mathop \to c^-} \map f x$ and $\ds \lim_{x \mathop \to c^+} \map f x$ are [[Definition:Real Number|finite]].
Since $f$ is [[Definition:Increasing Real Function|increasing]]:
:$\ds \lim_{x \mathop \to c^-} \map f x < \lim_{x \mathop \to c... | Monotone Real Function with Everywhere Dense Image is Continuous/Lemma | https://proofwiki.org/wiki/Monotone_Real_Function_with_Everywhere_Dense_Image_is_Continuous/Lemma | https://proofwiki.org/wiki/Monotone_Real_Function_with_Everywhere_Dense_Image_is_Continuous/Lemma | [
"Real Analysis"
] | [] | [
"Discontinuity of Monotonic Function is Jump Discontinuity",
"Definition:Real Number",
"Definition:Increasing/Real Function"
] |
proofwiki-11576 | Defining Sequence of Natural Logarithm is Uniformly Convergent on Compact Sets | Let $x \in \R$ be a real number such that $x > 0$.
Let $\sequence {f_n}$ be the sequence of mappings $f_n : \R_{>0} \to \R$ defined as:
:$\map {f_n} x = n \paren {\sqrt [n] x - 1}$
Let $K \subseteq \R_{>0}$ be compact.
Then $\sequence {f_n}$ is uniformly convergent on $K$. | From Continuity of Root Function and Combination Theorem for Continuous Real Functions:
:$\forall n \in \N : \map {f_n} x$ is continuous on $K$
From Defining Sequence of Natural Logarithm is Convergent, $\sequence {f_n}$ is pointwise convergent to $\ln$.
From Real Natural Logarithm Function is Continuous, $\ln$ is cont... | Let $x \in \R$ be a [[Definition:Real Number|real number]] such that $x > 0$.
Let $\sequence {f_n}$ be the [[Definition:Sequence|sequence]] of mappings $f_n : \R_{>0} \to \R$ defined as:
:$\map {f_n} x = n \paren {\sqrt [n] x - 1}$
Let $K \subseteq \R_{>0}$ be [[Definition:Compact Subset of Real Numbers|compact]].
... | From [[Continuity of Root Function]] and [[Combination Theorem for Continuous Real Functions]]:
:$\forall n \in \N : \map {f_n} x$ is [[Definition:Continuous Real Function|continuous]] on $K$
From [[Defining Sequence of Natural Logarithm is Convergent]], $\sequence {f_n}$ is [[Definition:Pointwise Convergence|pointwis... | Defining Sequence of Natural Logarithm is Uniformly Convergent on Compact Sets | https://proofwiki.org/wiki/Defining_Sequence_of_Natural_Logarithm_is_Uniformly_Convergent_on_Compact_Sets | https://proofwiki.org/wiki/Defining_Sequence_of_Natural_Logarithm_is_Uniformly_Convergent_on_Compact_Sets | [
"Natural Logarithms"
] | [
"Definition:Real Number",
"Definition:Sequence",
"Definition:Compact Space/Real Analysis",
"Definition:Uniform Convergence"
] | [
"Continuity of Root Function",
"Combination Theorem for Continuous Functions/Real",
"Definition:Continuous Real Function",
"Defining Sequence of Natural Logarithm is Convergent",
"Definition:Pointwise Convergence",
"Real Natural Logarithm Function is Continuous",
"Defining Sequence of Natural Logarithm ... |
proofwiki-11577 | Multiplication of Positive Number by Real Number Greater than One | Let $x$ and $y$ be real numbers.
Let $x > 1$.
Let $y > 0$.
Then $\dfrac y x < y$. | {{begin-eqn}}
{{eqn | l = x
| o = <
| r = 1
}}
{{eqn | ll= \leadsto
| l = \frac 1 x
| o = <
| r = 1
| c = Ordering of Reciprocals
}}
{{eqn | ll= \leadsto
| l = \frac y x
| o = <
| r = y
| c = Real Number Ordering is Compatible with Multiplication
}}
{{end-eqn}... | Let $x$ and $y$ be [[Definition:Real Number|real numbers]].
Let $x > 1$.
Let $y > 0$.
Then $\dfrac y x < y$. | {{begin-eqn}}
{{eqn | l = x
| o = <
| r = 1
}}
{{eqn | ll= \leadsto
| l = \frac 1 x
| o = <
| r = 1
| c = [[Ordering of Reciprocals]]
}}
{{eqn | ll= \leadsto
| l = \frac y x
| o = <
| r = y
| c = [[Real Number Ordering is Compatible with Multiplication]]
}}
{{... | Multiplication of Positive Number by Real Number Greater than One | https://proofwiki.org/wiki/Multiplication_of_Positive_Number_by_Real_Number_Greater_than_One | https://proofwiki.org/wiki/Multiplication_of_Positive_Number_by_Real_Number_Greater_than_One | [
"Real Multiplication",
"Inequalities"
] | [
"Definition:Real Number"
] | [
"Ordering of Reciprocals",
"Real Number Ordering is Compatible with Multiplication",
"Category:Real Multiplication",
"Category:Inequalities"
] |
proofwiki-11578 | Set of Finite Suprema is Directed | Let $\struct {S, \vee, \preceq}$ be a join semilattice.
Let $X$ be a non-empty subset of $S$.
Then
:$\set {\sup A: A \in \map {\operatorname {Fin} } X \land A \ne \O}$ is directed.
where $\map {\operatorname {Fin} } X$ denotes the set of all finite subsets of $X$. | By Existence of Non-Empty Finite Suprema in Join Semilattice:
:for every $A \in \map {\operatorname {Fin} } S$ if $A \ne \O$, then $A$ admits a supremum.
By definition of non-empty set:
:$\exists a: a \in X$
By definitions of subset and singleton:
:$\set x \subseteq X$
By Singleton is Finite:
:$\set x$ is finite
By def... | Let $\struct {S, \vee, \preceq}$ be a [[Definition:Join Semilattice|join semilattice]].
Let $X$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of $S$.
Then
:$\set {\sup A: A \in \map {\operatorname {Fin} } X \land A \ne \O}$ is [[Definition:Directed Subset|directed]].
where $\map {\operator... | By [[Existence of Non-Empty Finite Suprema in Join Semilattice]]:
:for every $A \in \map {\operatorname {Fin} } S$ if $A \ne \O$, then $A$ admits a [[Definition:Supremum of Set|supremum]].
By definition of [[Definition:Non-Empty Set|non-empty set]]:
:$\exists a: a \in X$
By definitions of [[Definition:Subset|subset]]... | Set of Finite Suprema is Directed | https://proofwiki.org/wiki/Set_of_Finite_Suprema_is_Directed | https://proofwiki.org/wiki/Set_of_Finite_Suprema_is_Directed | [
"Join and Meet Semilattices"
] | [
"Definition:Join Semilattice",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Directed Subset",
"Definition:Set of Sets",
"Definition:Finite Set",
"Definition:Subset"
] | [
"Existence of Non-Empty Finite Suprema in Join Semilattice",
"Definition:Supremum of Set",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Singleton",
"Singleton is Finite",
"Definition:Finite Set",
"Definition:Non-Empty Set",
"Definition:Singleton",
"Definition:Non-Empty Set",
"Fin... |
proofwiki-11579 | Change of Index Variable of Summation | :$\ds \sum_{\map R i} a_i = \sum_{\map R j} a_j$ | {{ProofWanted|Too obvious?}} | :$\ds \sum_{\map R i} a_i = \sum_{\map R j} a_j$ | {{ProofWanted|Too obvious?}} | Change of Index Variable of Summation | https://proofwiki.org/wiki/Change_of_Index_Variable_of_Summation | https://proofwiki.org/wiki/Change_of_Index_Variable_of_Summation | [
"Summations"
] | [] | [] |
proofwiki-11580 | Singleton is Finite | Let $x$ be arbitrary.
Then $\set x$ is a finite set. | Define a mapping $f: \set x \to \N_{< 1}$:
:$\map f x = 0$
By definition of singleton:
:$\forall y, z \in \set x: \map f y = \map f z \implies y = z$
By definition:
:$f$ is an injection.
By definition of initial segment of natural numbers:
:$\N_{< 1} = \set 0$
By definition of $f$:
:$\forall n \in N_{< 1}: \exists z \i... | Let $x$ be arbitrary.
Then $\set x$ is a [[Definition:Finite Set|finite set]]. | Define a [[Definition:Mapping|mapping]] $f: \set x \to \N_{< 1}$:
:$\map f x = 0$
By definition of [[Definition:Singleton|singleton]]:
:$\forall y, z \in \set x: \map f y = \map f z \implies y = z$
By definition:
:$f$ is an [[Definition:Injection|injection]].
By definition of [[Definition:Initial Segment of Natural ... | Singleton is Finite | https://proofwiki.org/wiki/Singleton_is_Finite | https://proofwiki.org/wiki/Singleton_is_Finite | [
"Singletons"
] | [
"Definition:Finite Set"
] | [
"Definition:Mapping",
"Definition:Singleton",
"Definition:Injection",
"Definition:Initial Segment of Natural Numbers",
"Definition:Surjection",
"Definition:Bijection",
"Definition:Set Equivalence",
"Definition:Finite Set"
] |
proofwiki-11581 | Normal Property is Weakly Hereditary | If $T$ is a normal space then $T_K$ is also a normal space.
That is, the property of being a normal space is weakly hereditary. | {{Recall|Normal Space|normal space|index = 1}}
{{:Definition:Normal Space/Definition 1}}
From $T_1$ Property is Hereditary, an arbitrary subspace of a $T_1$ space is also a $T_1$ space.
From $T_4$ Property is Weakly Hereditary, an arbitrary closed subspace of a $T_4$ space is also a $T_4$ space.
Hence the result.
{{qed... | If $T$ is a [[Definition:Normal Space|normal space]] then $T_K$ is also a [[Definition:Normal Space|normal space]].
That is, the property of being a [[Definition:Normal Space|normal space]] is [[Definition:Weakly Hereditary Property|weakly hereditary]]. | {{Recall|Normal Space|normal space|index = 1}}
{{:Definition:Normal Space/Definition 1}}
From [[T1 Property is Hereditary|$T_1$ Property is Hereditary]], an arbitrary [[Definition:Topological Subspace|subspace]] of a [[Definition:T1 Space|$T_1$ space]] is also a [[Definition:T1 Space|$T_1$ space]].
From [[T4 Property... | Normal Property is Weakly Hereditary | https://proofwiki.org/wiki/Normal_Property_is_Weakly_Hereditary | https://proofwiki.org/wiki/Normal_Property_is_Weakly_Hereditary | [
"Normal Spaces",
"Separation Properties Preserved in Subspace",
"Examples of Weakly Hereditary Properties"
] | [
"Definition:Normal Space",
"Definition:Normal Space",
"Definition:Normal Space",
"Definition:Weakly Hereditary Property"
] | [
"T1 Property is Hereditary",
"Definition:Topological Subspace",
"Definition:T1 Space",
"Definition:T1 Space",
"T4 Property is Weakly Hereditary",
"Definition:Closed Set/Topology",
"Definition:Topological Subspace",
"Definition:T4 Space",
"Definition:T4 Space"
] |
proofwiki-11582 | Derivative of Natural Logarithm Function/Proof 4/Lemma | Let $\sequence {f_n}_n$ be the sequence of real functions $f_n: \R_{>0} \to \R$ defined as:
:$\map {f_n} x = n \paren {\sqrt [n] x - 1}$
Let $k \in \N$.
Let $J = \closedint {\dfrac 1 k} k$.
Then the sequence of derivatives $\sequence { {f_n}'}_n$ converges uniformly to some real function $g: J \to \R$. | From Derivative of $n$th Root and Combination Theorem for Sequences:
:$\forall n \in \N: \forall x \in J : D_x \map {f_n} x = \dfrac {\sqrt [n] x} x$
From Closed Bounded Subset of Real Numbers is Compact, $J$ is compact.
Thus:
{{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} D_x \map {f_n} x
| r = \lim_{n \m... | Let $\sequence {f_n}_n$ be the [[Definition:Sequence|sequence]] of [[Definition:Real Function|real functions]] $f_n: \R_{>0} \to \R$ defined as:
:$\map {f_n} x = n \paren {\sqrt [n] x - 1}$
Let $k \in \N$.
Let $J = \closedint {\dfrac 1 k} k$.
Then the [[Definition:Sequence|sequence]] of [[Definition:Derivative of R... | From [[Derivative of Nth Root|Derivative of $n$th Root]] and [[Combination Theorem for Sequences]]:
:$\forall n \in \N: \forall x \in J : D_x \map {f_n} x = \dfrac {\sqrt [n] x} x$
From [[Closed Bounded Subset of Real Numbers is Compact]], $J$ is [[Definition:Compact (Real Analysis)|compact]].
Thus:
{{begin-eqn}}
... | Derivative of Natural Logarithm Function/Proof 4/Lemma | https://proofwiki.org/wiki/Derivative_of_Natural_Logarithm_Function/Proof_4/Lemma | https://proofwiki.org/wiki/Derivative_of_Natural_Logarithm_Function/Proof_4/Lemma | [
"Derivative of Natural Logarithm Function"
] | [
"Definition:Sequence",
"Definition:Real Function",
"Definition:Sequence",
"Definition:Derivative/Real Function",
"Definition:Uniform Convergence",
"Definition:Real Function"
] | [
"Derivative of Nth Root",
"Combination Theorem for Sequences",
"Closed Bounded Subset of Real Numbers is Compact",
"Definition:Compact Space/Real Analysis",
"Combination Theorem for Sequences/Real/Multiple Rule",
"Power Function tends to One as Power tends to Zero/Rational Number",
"Definition:Pointwise... |
proofwiki-11583 | Exponential Function is Well-Defined/Real | Let $x \in \R$ be a real number.
Let $\exp x$ be the exponential of $x$.
Then $\exp x$ is well-defined. | This proof assumes the power series definition of $\exp$.
From Series of Power over Factorial Converges:
:$\ds \sum_{n \mathop = 0}^\infty \frac {x^n} {n!}$ converges
Hence the result, from Convergent Real Sequence has Unique Limit.
{{qed}} | Let $x \in \R$ be a [[Definition:Real Number|real number]].
Let $\exp x$ be the [[Definition:Real Exponential Function|exponential of $x$]].
Then $\exp x$ is [[Definition:Well-Defined Mapping|well-defined]]. | This proof assumes the [[Definition:Exponential Function/Real/Power Series Expansion|power series definition of $\exp$]].
From [[Series of Power over Factorial Converges]]:
:$\ds \sum_{n \mathop = 0}^\infty \frac {x^n} {n!}$ [[Definition:Convergent Series|converges]]
Hence the result, from [[Convergent Real Sequence... | Exponential Function is Well-Defined/Real/Proof 1 | https://proofwiki.org/wiki/Exponential_Function_is_Well-Defined/Real | https://proofwiki.org/wiki/Exponential_Function_is_Well-Defined/Real/Proof_1 | [
"Exponential Function",
"Exponential Function is Well-Defined"
] | [
"Definition:Real Number",
"Definition:Exponential Function/Real",
"Definition:Well-Defined/Mapping"
] | [
"Definition:Exponential Function/Real/Power Series Expansion",
"Series of Power over Factorial Converges",
"Definition:Convergent Series",
"Convergent Real Sequence has Unique Limit"
] |
proofwiki-11584 | Exponential Function is Well-Defined/Real | Let $x \in \R$ be a real number.
Let $\exp x$ be the exponential of $x$.
Then $\exp x$ is well-defined. | This proof assumes the sequence definition of $\exp$.
Let $\sequence {f_n}$ be the sequence of mappings $f_n : \R \to \R$ defined as:
:$\map {f_n} x = \paren {1 + \dfrac x n}^n$
Fix $x \in \R$.
Then:
{{begin-eqn}}
{{eqn | l = \map {f_n} x
| r = \paren {1 + \dfrac x n}^n
| c = Definition of $\map {f_n} x$
}}... | Let $x \in \R$ be a [[Definition:Real Number|real number]].
Let $\exp x$ be the [[Definition:Real Exponential Function|exponential of $x$]].
Then $\exp x$ is [[Definition:Well-Defined Mapping|well-defined]]. | This proof assumes the [[Definition:Exponential Function/Real/Limit of Sequence|sequence definition of $\exp$]].
Let $\sequence {f_n}$ be the [[Definition:Sequence|sequence]] of mappings $f_n : \R \to \R$ defined as:
:$\map {f_n} x = \paren {1 + \dfrac x n}^n$
Fix $x \in \R$.
Then:
{{begin-eqn}}
{{eqn | l = \map {f_... | Exponential Function is Well-Defined/Real/Proof 2 | https://proofwiki.org/wiki/Exponential_Function_is_Well-Defined/Real | https://proofwiki.org/wiki/Exponential_Function_is_Well-Defined/Real/Proof_2 | [
"Exponential Function",
"Exponential Function is Well-Defined"
] | [
"Definition:Real Number",
"Definition:Exponential Function/Real",
"Definition:Well-Defined/Mapping"
] | [
"Definition:Exponential Function/Real/Limit of Sequence",
"Definition:Sequence",
"Binomial Theorem/Integral Index",
"Negative of Absolute Value",
"Absolute Value Function is Completely Multiplicative",
"Multiplication of Positive Number by Real Number Greater than One",
"Series of Power over Factorial C... |
proofwiki-11585 | Exponential Function is Well-Defined/Real | Let $x \in \R$ be a real number.
Let $\exp x$ be the exponential of $x$.
Then $\exp x$ is well-defined. | This proof assumes the continuous extension definition of $\exp$.
Let $e$ denote Euler's number.
Let $f: \Q \to \R$ be the real-valued function defined as:
:$f \left({r}\right) = e^r$
From Euler's Number to Rational Power permits Unique Continuous Extension, there exists a unique continuous extension of $f$ to $\R$.
He... | Let $x \in \R$ be a [[Definition:Real Number|real number]].
Let $\exp x$ be the [[Definition:Real Exponential Function|exponential of $x$]].
Then $\exp x$ is [[Definition:Well-Defined Mapping|well-defined]]. | This proof assumes the [[Definition:Exponential Function/Real/Extension of Rational Exponential|continuous extension definition of $\exp$]].
Let $e$ denote [[Definition:Euler's Number|Euler's number]].
Let $f: \Q \to \R$ be the [[Definition:Real-Valued Function|real-valued function]] defined as:
:$f \left({r}\right) ... | Exponential Function is Well-Defined/Real/Proof 3 | https://proofwiki.org/wiki/Exponential_Function_is_Well-Defined/Real | https://proofwiki.org/wiki/Exponential_Function_is_Well-Defined/Real/Proof_3 | [
"Exponential Function",
"Exponential Function is Well-Defined"
] | [
"Definition:Real Number",
"Definition:Exponential Function/Real",
"Definition:Well-Defined/Mapping"
] | [
"Definition:Exponential Function/Real/Extension of Rational Exponential",
"Definition:Euler's Number",
"Definition:Real-Valued Function",
"Euler's Number to Rational Power permits Unique Continuous Extension",
"Definition:Unique",
"Definition:Continuous Extension"
] |
proofwiki-11586 | Exponential Function is Well-Defined/Real | Let $x \in \R$ be a real number.
Let $\exp x$ be the exponential of $x$.
Then $\exp x$ is well-defined. | This proof assumes the definition of the exponential as the inverse of the logarithm.
From Logarithm is Strictly Increasing, $\ln$ is strictly monotone on $\R_{>0}$.
From Inverse of Strictly Monotone Function, $f$ permits an inverse mapping.
Hence the result, from Inverse Mapping is Unique.
{{qed}} | Let $x \in \R$ be a [[Definition:Real Number|real number]].
Let $\exp x$ be the [[Definition:Real Exponential Function|exponential of $x$]].
Then $\exp x$ is [[Definition:Well-Defined Mapping|well-defined]]. | This proof assumes the definition of the [[Definition:Exponential Function/Real/Inverse of Natural Logarithm|exponential as the inverse of the logarithm]].
From [[Logarithm is Strictly Increasing]], $\ln$ is [[Definition:Strictly Monotone Real Function|strictly monotone]] on $\R_{>0}$.
From [[Inverse of Strictly Mon... | Exponential Function is Well-Defined/Real/Proof 4 | https://proofwiki.org/wiki/Exponential_Function_is_Well-Defined/Real | https://proofwiki.org/wiki/Exponential_Function_is_Well-Defined/Real/Proof_4 | [
"Exponential Function",
"Exponential Function is Well-Defined"
] | [
"Definition:Real Number",
"Definition:Exponential Function/Real",
"Definition:Well-Defined/Mapping"
] | [
"Definition:Exponential Function/Real/Inverse of Natural Logarithm",
"Logarithm is Strictly Increasing",
"Definition:Strictly Monotone/Real Function",
"Inverse of Strictly Monotone Function",
"Definition:Inverse Mapping",
"Inverse Mapping is Unique"
] |
proofwiki-11587 | Exponential Function is Well-Defined/Real | Let $x \in \R$ be a real number.
Let $\exp x$ be the exponential of $x$.
Then $\exp x$ is well-defined. | This proof assumes the definition of $\exp$ as the solution to an initial value problem.
That is, suppose $\exp$ solves:
:$(1): \quad \dfrac \d {\d x} y = \map f {x, y}$
:$(2): \quad \map \exp 0 = 1$
on $\R$, where $\map f {x, y} = y$.
From Derivative of Exponential Function: Proof 4, the function $\phi : \R \to \R$ de... | Let $x \in \R$ be a [[Definition:Real Number|real number]].
Let $\exp x$ be the [[Definition:Real Exponential Function|exponential of $x$]].
Then $\exp x$ is [[Definition:Well-Defined Mapping|well-defined]]. | This proof assumes the [[Definition:Exponential Function/Real/Differential Equation|definition of $\exp$ as the solution to an initial value problem]].
That is, suppose $\exp$ solves:
:$(1): \quad \dfrac \d {\d x} y = \map f {x, y}$
:$(2): \quad \map \exp 0 = 1$
on $\R$, where $\map f {x, y} = y$.
From [[Derivative ... | Exponential Function is Well-Defined/Real/Proof 5 | https://proofwiki.org/wiki/Exponential_Function_is_Well-Defined/Real | https://proofwiki.org/wiki/Exponential_Function_is_Well-Defined/Real/Proof_5 | [
"Exponential Function",
"Exponential Function is Well-Defined"
] | [
"Definition:Real Number",
"Definition:Exponential Function/Real",
"Definition:Well-Defined/Mapping"
] | [
"Definition:Exponential Function/Real/Differential Equation",
"Derivative of Exponential Function/Proof 4",
"Exponential of Zero/Proof 3",
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Initial Value Problem",
"Exponential Function is Continuous/Real Numbers/Proof 5",
"Defi... |
proofwiki-11588 | Tail of Convergent Sequence | Let $\left\langle{a_n}\right\rangle$ be a real sequence.
Let $m \in \N$ be a natural number.
Let $a \in \R$ be a real number.
Then:
:$a_n \to a$
{{iff}}:
:$a_{n + m} \to a$ | === Necessary Condition ===
Suppose that $a_n \to a$.
Then for each $\epsilon > 0$, there exists $N \in \N$ such that:
:$\size {a_n - a} < \epsilon$ for $n \ge N$.
Set:
:$N^\ast = \max \set {1, N - m}$
Then for $n \ge N^\ast$, we have $n \ge N - m$ and so $n + m \ge N$.
Then:
:$\size {a_{n + m} - a} < \epsilon$ f... | Let $\left\langle{a_n}\right\rangle$ be a [[Definition:Real Sequence|real sequence]].
Let $m \in \N$ be a [[Definition:Natural Numbers|natural number]].
Let $a \in \R$ be a [[Definition:Real Number|real number]].
Then:
:$a_n \to a$
{{iff}}:
:$a_{n + m} \to a$ | === Necessary Condition ===
Suppose that $a_n \to a$.
Then for each $\epsilon > 0$, there exists $N \in \N$ such that:
:$\size {a_n - a} < \epsilon$ for $n \ge N$.
Set:
:$N^\ast = \max \set {1, N - m}$
Then for $n \ge N^\ast$, we have $n \ge N - m$ and so $n + m \ge N$.
Then:
:$\size {a_{n + m} - a} < \ep... | Tail of Convergent Sequence | https://proofwiki.org/wiki/Tail_of_Convergent_Sequence | https://proofwiki.org/wiki/Tail_of_Convergent_Sequence | [
"Convergence",
"Sequences"
] | [
"Definition:Real Sequence",
"Definition:Natural Numbers",
"Definition:Real Number"
] | [] |
proofwiki-11589 | Union of Inverses of Mappings is Inverse of Union of Mappings | Let $I$ be an indexing set.
Let $\family {f_i: i \in I}$ be an indexed family of mappings.
For each $i \in I$, let $f^{-1}$ denote the inverse of $f$.
Then the inverse of the union of $\family {f_i: i \in I}$ is the union of the inverses of $f_i, i \in I$.
That is:
:$\ds \paren {\bigcup \family {f_i: i \in I} }^{-1} = ... | {{begin-eqn}}
{{eqn | l = \tuple {y, x}
| o = \in
| r = \paren {\bigcup \family {f_i: i \in I} }^{-1}
}}
{{eqn | ll= \leadstoandfrom
| l = \tuple {x, y}
| o = \in
| r = \family {f_i: i \in I}
| c = {{Defof|Inverse Relation}}
}}
{{eqn | ll= \leadstoandfrom
| q = \exists i \in I
... | Let $I$ be an [[Definition:Indexing Set|indexing set]].
Let $\family {f_i: i \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Mapping|mappings]].
For each $i \in I$, let $f^{-1}$ denote the [[Definition:Inverse of Mapping|inverse of $f$]].
Then the [[Definition:Inverse Relation|inverse]] o... | {{begin-eqn}}
{{eqn | l = \tuple {y, x}
| o = \in
| r = \paren {\bigcup \family {f_i: i \in I} }^{-1}
}}
{{eqn | ll= \leadstoandfrom
| l = \tuple {x, y}
| o = \in
| r = \family {f_i: i \in I}
| c = {{Defof|Inverse Relation}}
}}
{{eqn | ll= \leadstoandfrom
| q = \exists i \in I
... | Union of Inverses of Mappings is Inverse of Union of Mappings | https://proofwiki.org/wiki/Union_of_Inverses_of_Mappings_is_Inverse_of_Union_of_Mappings | https://proofwiki.org/wiki/Union_of_Inverses_of_Mappings_is_Inverse_of_Union_of_Mappings | [
"Inverse Relations",
"Set Union",
"Indexed Families"
] | [
"Definition:Indexing Set",
"Definition:Indexing Set/Family",
"Definition:Mapping",
"Definition:Inverse of Mapping",
"Definition:Inverse Relation",
"Definition:Set Union/Family of Sets",
"Definition:Set Union/Family of Sets",
"Definition:Inverse Relation"
] | [
"Category:Inverse Relations",
"Category:Set Union",
"Category:Indexed Families"
] |
proofwiki-11590 | Union of Functions Theorem/Corollary | For each $i \in \N$, let $g_i : X_i \to Y$ be invertible.
Then $\ds \bigcup \set {g_i: i \in \N}$ is invertible and:
:$\ds \paren {\bigcup \set {g_i: i \in \N} }^{-1} = \bigcup \set {g_i^{-1}: i \in \N}$ | {{ProofWanted}}
Category:Inverse Mappings
943qyuq5ivgtxh7mueilwt404u2vfoh | For each $i \in \N$, let $g_i : X_i \to Y$ be [[Definition:Inverse Mapping|invertible]].
Then $\ds \bigcup \set {g_i: i \in \N}$ is [[Definition:Inverse Mapping|invertible]] and:
:$\ds \paren {\bigcup \set {g_i: i \in \N} }^{-1} = \bigcup \set {g_i^{-1}: i \in \N}$ | {{ProofWanted}}
[[Category:Inverse Mappings]]
943qyuq5ivgtxh7mueilwt404u2vfoh | Union of Functions Theorem/Corollary | https://proofwiki.org/wiki/Union_of_Functions_Theorem/Corollary | https://proofwiki.org/wiki/Union_of_Functions_Theorem/Corollary | [
"Inverse Mappings"
] | [
"Definition:Inverse Mapping",
"Definition:Inverse Mapping"
] | [
"Category:Inverse Mappings"
] |
proofwiki-11591 | Permutation of Indices of Summation | :$\ds \sum_{\map R j} a_j = \sum_{\map R {\map \pi j} } a_{\map \pi j}$ | {{begin-eqn}}
{{eqn | l = \sum_{\map R {\map \pi j} } a_{\map \pi j}
| r = \sum_{j \mathop \in \Z} a_{\map \pi j} \sqbrk {\map R {\map \pi j} }
| c = {{Defof|Summation by Iverson's Convention}}
}}
{{eqn | r = \sum_{j \mathop \in \Z} \sum_{i \mathop \in \Z} a_i \sqbrk {\map R i} \sqbrk {i = \map \pi j}
... | :$\ds \sum_{\map R j} a_j = \sum_{\map R {\map \pi j} } a_{\map \pi j}$ | {{begin-eqn}}
{{eqn | l = \sum_{\map R {\map \pi j} } a_{\map \pi j}
| r = \sum_{j \mathop \in \Z} a_{\map \pi j} \sqbrk {\map R {\map \pi j} }
| c = {{Defof|Summation by Iverson's Convention}}
}}
{{eqn | r = \sum_{j \mathop \in \Z} \sum_{i \mathop \in \Z} a_i \sqbrk {\map R i} \sqbrk {i = \map \pi j}
... | Permutation of Indices of Summation/Proof | https://proofwiki.org/wiki/Permutation_of_Indices_of_Summation | https://proofwiki.org/wiki/Permutation_of_Indices_of_Summation/Proof | [
"Summations"
] | [] | [
"Change of Index Variable of Summation"
] |
proofwiki-11592 | Translation of Index Variable of Summation | Let $R: \Z \to \set {\T, \F}$ be a propositional function on the set of integers.
Let $\ds \sum_{\map R j} a_j$ denote a summation over $R$.
Then:
:$\ds \sum_{\map R j} a_j = \sum_{\map R {c \mathop + j} } a_{c \mathop + j} = \sum_{\map R {c \mathop - j} } a_{c \mathop - j}$
where $c$ is an integer constant which is no... | First we consider the case where the fiber of truth of $R$ is finite.
{{ProofWanted}} | Let $R: \Z \to \set {\T, \F}$ be a [[Definition:Propositional Function|propositional function]] on the set of [[Definition:Integer|integers]].
Let $\ds \sum_{\map R j} a_j$ denote a [[Definition:Summation by Propositional Function|summation]] over $R$.
Then:
:$\ds \sum_{\map R j} a_j = \sum_{\map R {c \mathop + j} } ... | First we consider the case where the [[Definition:Fiber of Truth|fiber of truth]] of $R$ is [[Definition:Finite Set|finite]].
{{ProofWanted}} | Translation of Index Variable of Summation | https://proofwiki.org/wiki/Translation_of_Index_Variable_of_Summation | https://proofwiki.org/wiki/Translation_of_Index_Variable_of_Summation | [
"Summations"
] | [
"Definition:Propositional Function",
"Definition:Integer",
"Definition:Summation/Propositional Function",
"Definition:Integer",
"Definition:Constant"
] | [
"Definition:Fiber of Truth",
"Definition:Finite Set",
"Definition:Fiber of Truth"
] |
proofwiki-11593 | Axiom of Approximation in Up-Complete Semilattice | Let $\mathscr S = \struct {S, \wedge, \preceq}$ be an up-complete meet semilattice.
Let:
:$\forall x \in S: x^\ll$ is directed
Then:
:$\mathscr S$ satisfies the axiom of approximation
{{iff}}:
:$\forall x, y \in S: x \npreceq y \implies \exists u \in S: u \ll x \land u \npreceq y$ | === Sufficient Condition ===
Let $\mathscr S$ satisfy the axiom of approximation.
Let $x, y \in S$ such that
:$x \npreceq y$
By assumption:
:$x^\ll$ is directed.
By definition of up-complete:
:$x^\ll$ admits a supremum.
By the axiom of approximation:
:$x = \map \sup {x^\ll}$
By definition of supremum:
:if $y$ is upper ... | Let $\mathscr S = \struct {S, \wedge, \preceq}$ be an [[Definition:Up-Complete|up-complete]] [[Definition:Meet Semilattice|meet semilattice]].
Let:
:$\forall x \in S: x^\ll$ is [[Definition:Directed Subset|directed]]
Then:
:$\mathscr S$ satisfies the [[Axiom:Axiom of Approximation|axiom of approximation]]
{{iff}}:
:... | === Sufficient Condition ===
Let $\mathscr S$ satisfy the [[Axiom:Axiom of Approximation|axiom of approximation]].
Let $x, y \in S$ such that
:$x \npreceq y$
By assumption:
:$x^\ll$ is [[Definition:Directed Subset|directed]].
By definition of [[Definition:Up-Complete|up-complete]]:
:$x^\ll$ admits a [[Definition:Su... | Axiom of Approximation in Up-Complete Semilattice | https://proofwiki.org/wiki/Axiom_of_Approximation_in_Up-Complete_Semilattice | https://proofwiki.org/wiki/Axiom_of_Approximation_in_Up-Complete_Semilattice | [
"Way Below Relation",
"Up-Complete Semilattices"
] | [
"Definition:Up-Complete",
"Definition:Meet Semilattice",
"Definition:Directed Subset",
"Axiom:Axiom of Approximation"
] | [
"Axiom:Axiom of Approximation",
"Definition:Directed Subset",
"Definition:Up-Complete",
"Definition:Supremum of Set",
"Axiom:Axiom of Approximation",
"Definition:Supremum of Set",
"Definition:Upper Bound of Set",
"Definition:Upper Bound of Set",
"Definition:Way Below Closure",
"Definition:Directed... |
proofwiki-11594 | Operand is Upper Bound of Way Below Closure | Let $\struct {S, \preceq}$ be an ordered set.
Let $x \in S$.
Then
:$x$ is upper bound for $x^\ll$
where $x^\ll$ denotes the way below closure of $x$. | Let $y \in x^\ll$
By definition of way below closure:
:$y \ll x$
where $\ll$ denotes the way below relation.
Thus by Way Below implies Preceding:
:$y \preceq x$
Thus by definition:
:$x$ is upper bound for $x^\ll$
{{qed}} | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $x \in S$.
Then
:$x$ is [[Definition:Upper Bound of Set|upper bound]] for $x^\ll$
where $x^\ll$ denotes the [[Definition:Way Below Closure|way below closure]] of $x$. | Let $y \in x^\ll$
By definition of [[Definition:Way Below Closure|way below closure]]:
:$y \ll x$
where $\ll$ denotes the [[Definition:Element is Way Below|way below relation]].
Thus by [[Way Below implies Preceding]]:
:$y \preceq x$
Thus by definition:
:$x$ is [[Definition:Upper Bound of Set|upper bound]] for $x^\l... | Operand is Upper Bound of Way Below Closure | https://proofwiki.org/wiki/Operand_is_Upper_Bound_of_Way_Below_Closure | https://proofwiki.org/wiki/Operand_is_Upper_Bound_of_Way_Below_Closure | [
"Way Below Relation"
] | [
"Definition:Ordered Set",
"Definition:Upper Bound of Set",
"Definition:Way Below Closure"
] | [
"Definition:Way Below Closure",
"Definition:Element is Way Below",
"Way Below implies Preceding",
"Definition:Upper Bound of Set"
] |
proofwiki-11595 | Exchange of Order of Summation | :$\ds \sum_{\map R i} \sum_{\map S j} a_{i j} = \sum_{\map S j} \sum_{\map R i} a_{i j}$ | {{begin-eqn}}
{{eqn | l = \sum_{\map R i} \sum_{\map S {i, j} } a_{i j}
| r = \sum_{i, j \mathop \in \Z} a_{i j} \sqbrk {\map R i} \sqbrk {\map S {i, j} }
| c =
}}
{{eqn | r = \sum_{i, j \mathop \in \Z} a_{i j} \sqbrk {\map R i \land \map S {i, j} }
| c =
}}
{{eqn | r = \sum_{i, j \mathop \in \Z} a_... | :$\ds \sum_{\map R i} \sum_{\map S j} a_{i j} = \sum_{\map S j} \sum_{\map R i} a_{i j}$ | {{begin-eqn}}
{{eqn | l = \sum_{\map R i} \sum_{\map S {i, j} } a_{i j}
| r = \sum_{i, j \mathop \in \Z} a_{i j} \sqbrk {\map R i} \sqbrk {\map S {i, j} }
| c =
}}
{{eqn | r = \sum_{i, j \mathop \in \Z} a_{i j} \sqbrk {\map R i \land \map S {i, j} }
| c =
}}
{{eqn | r = \sum_{i, j \mathop \in \Z} a_... | Exchange of Order of Summation with Dependency on Both Indices/Proof | https://proofwiki.org/wiki/Exchange_of_Order_of_Summation | https://proofwiki.org/wiki/Exchange_of_Order_of_Summation_with_Dependency_on_Both_Indices/Proof | [
"Summations"
] | [] | [] |
proofwiki-11596 | Exchange of Order of Summation | :$\ds \sum_{\map R i} \sum_{\map S j} a_{i j} = \sum_{\map S j} \sum_{\map R i} a_{i j}$ | Let $n$ be the cardinality of $T$.
The proof goes by induction on $n$.
=== Basis for the Induction ===
Let $n = 0$.
{{finish}}
=== Induction Step ===
Let $n > 0$.
Let $t \in T$.
Use Cardinality of Set minus Singleton
{{ProofWanted}} | :$\ds \sum_{\map R i} \sum_{\map S j} a_{i j} = \sum_{\map S j} \sum_{\map R i} a_{i j}$ | Let $n$ be the [[Definition:Cardinality of Finite Set|cardinality]] of $T$.
The proof goes by [[Principle of Mathematical Induction|induction]] on $n$.
=== Basis for the Induction ===
Let $n = 0$.
{{finish}}
=== Induction Step ===
Let $n > 0$.
Let $t \in T$.
Use [[Cardinality of Set minus Singleton]]
{{Proof... | Exchange of Order of Summations over Finite Sets/Cartesian Product/Proof 3 | https://proofwiki.org/wiki/Exchange_of_Order_of_Summation | https://proofwiki.org/wiki/Exchange_of_Order_of_Summations_over_Finite_Sets/Cartesian_Product/Proof_3 | [
"Summations"
] | [] | [
"Definition:Cardinality/Finite",
"Principle of Mathematical Induction",
"Cardinality of Set minus Singleton"
] |
proofwiki-11597 | Sum of Summations equals Summation of Sum/Infinite Sequence | Let $R: \Z \to \set {\T, \F}$ be a propositional function on the set of integers $\Z$.
Let $\ds \sum_{\map R i} x_i$ denote a summation over $R$.
Let the fiber of truth of $R$ be infinite.
Let $\ds \sum_{\map R i} b_i$ and $\ds \sum_{\map R i} c_i$ be convergent.
Then:
:$\ds \sum_{\map R i} \paren {b_i + c_i} = \sum_{\... | Let $b_i =: a_{i 1}$ and $c_i =: a_{i 2}$.
Then:
{{begin-eqn}}
{{eqn | l = \sum_{\map R i} \paren {b_i + c_i}
| r = \sum_{\map R i} \paren {a_{i 1} + a_{i 2} }
| c = by definition
}}
{{eqn | r = \sum_{\map R i} \paren {\sum_{1 \mathop \le j \mathop \le 2} a_{i j} }
| c = {{Defof|Summation}}
}}
{{eqn |... | Let $R: \Z \to \set {\T, \F}$ be a [[Definition:Propositional Function|propositional function]] on the [[Definition:Set|set]] of [[Definition:Integer|integers]] $\Z$.
Let $\ds \sum_{\map R i} x_i$ denote a [[Definition:Summation by Propositional Function|summation]] over $R$.
Let the [[Definition:Fiber of Truth|fibe... | Let $b_i =: a_{i 1}$ and $c_i =: a_{i 2}$.
Then:
{{begin-eqn}}
{{eqn | l = \sum_{\map R i} \paren {b_i + c_i}
| r = \sum_{\map R i} \paren {a_{i 1} + a_{i 2} }
| c = by definition
}}
{{eqn | r = \sum_{\map R i} \paren {\sum_{1 \mathop \le j \mathop \le 2} a_{i j} }
| c = {{Defof|Summation}}
}}
{{eqn... | Sum of Summations equals Summation of Sum/Infinite Sequence/Proof 1 | https://proofwiki.org/wiki/Sum_of_Summations_equals_Summation_of_Sum/Infinite_Sequence | https://proofwiki.org/wiki/Sum_of_Summations_equals_Summation_of_Sum/Infinite_Sequence/Proof_1 | [
"Summations"
] | [
"Definition:Propositional Function",
"Definition:Set",
"Definition:Integer",
"Definition:Summation/Propositional Function",
"Definition:Fiber of Truth",
"Definition:Infinite Set",
"Definition:Convergent Sequence/Real Numbers"
] | [
"Exchange of Order of Summation/Finite and Infinite Series"
] |
proofwiki-11598 | Sum of Summations equals Summation of Sum/Infinite Sequence | Let $R: \Z \to \set {\T, \F}$ be a propositional function on the set of integers $\Z$.
Let $\ds \sum_{\map R i} x_i$ denote a summation over $R$.
Let the fiber of truth of $R$ be infinite.
Let $\ds \sum_{\map R i} b_i$ and $\ds \sum_{\map R i} c_i$ be convergent.
Then:
:$\ds \sum_{\map R i} \paren {b_i + c_i} = \sum_{\... | By definition, $\ds \sum_{\map R i} b_i$ and $\ds \sum_{\map R i} c_i$ are sequences in $\R$.
Hence the result as an instance of Sum Rule for Real Sequences.
{{qed}} | Let $R: \Z \to \set {\T, \F}$ be a [[Definition:Propositional Function|propositional function]] on the [[Definition:Set|set]] of [[Definition:Integer|integers]] $\Z$.
Let $\ds \sum_{\map R i} x_i$ denote a [[Definition:Summation by Propositional Function|summation]] over $R$.
Let the [[Definition:Fiber of Truth|fibe... | By definition, $\ds \sum_{\map R i} b_i$ and $\ds \sum_{\map R i} c_i$ are [[Definition:Real Sequence|sequences]] in $\R$.
Hence the result as an instance of [[Sum Rule for Real Sequences]].
{{qed}} | Sum of Summations equals Summation of Sum/Infinite Sequence/Proof 2 | https://proofwiki.org/wiki/Sum_of_Summations_equals_Summation_of_Sum/Infinite_Sequence | https://proofwiki.org/wiki/Sum_of_Summations_equals_Summation_of_Sum/Infinite_Sequence/Proof_2 | [
"Summations"
] | [
"Definition:Propositional Function",
"Definition:Set",
"Definition:Integer",
"Definition:Summation/Propositional Function",
"Definition:Fiber of Truth",
"Definition:Infinite Set",
"Definition:Convergent Sequence/Real Numbers"
] | [
"Definition:Real Sequence",
"Combination Theorem for Sequences/Real/Sum Rule"
] |
proofwiki-11599 | Exchange of Order of Summation with Dependency on Both Indices | :$\ds \sum_{\map R i} \sum_{\map S {i, j} } a_{i j} = \sum_{\map {S'} j} \sum_{\map {R'} {i, j} } a_{i j}$
where:
:$\map {S'} j$ denotes the propositional function:
::there exists an $i$ such that both $\map R i$ and $\map S {i, j}$ hold
:$\map {R'} {i, j}$ denotes the propositional function:
::both $\map R i$ and $\ma... | {{begin-eqn}}
{{eqn | l = \sum_{\map R i} \sum_{\map S {i, j} } a_{i j}
| r = \sum_{i, j \mathop \in \Z} a_{i j} \sqbrk {\map R i} \sqbrk {\map S {i, j} }
| c =
}}
{{eqn | r = \sum_{i, j \mathop \in \Z} a_{i j} \sqbrk {\map R i \land \map S {i, j} }
| c =
}}
{{eqn | r = \sum_{i, j \mathop \in \Z} a_... | :$\ds \sum_{\map R i} \sum_{\map S {i, j} } a_{i j} = \sum_{\map {S'} j} \sum_{\map {R'} {i, j} } a_{i j}$
where:
:$\map {S'} j$ denotes the [[Definition:Propositional Function|propositional function]]:
::there exists an $i$ such that both $\map R i$ and $\map S {i, j}$ hold
:$\map {R'} {i, j}$ denotes the [[Definitio... | {{begin-eqn}}
{{eqn | l = \sum_{\map R i} \sum_{\map S {i, j} } a_{i j}
| r = \sum_{i, j \mathop \in \Z} a_{i j} \sqbrk {\map R i} \sqbrk {\map S {i, j} }
| c =
}}
{{eqn | r = \sum_{i, j \mathop \in \Z} a_{i j} \sqbrk {\map R i \land \map S {i, j} }
| c =
}}
{{eqn | r = \sum_{i, j \mathop \in \Z} a_... | Exchange of Order of Summation with Dependency on Both Indices/Proof | https://proofwiki.org/wiki/Exchange_of_Order_of_Summation_with_Dependency_on_Both_Indices | https://proofwiki.org/wiki/Exchange_of_Order_of_Summation_with_Dependency_on_Both_Indices/Proof | [
"Summations"
] | [
"Definition:Propositional Function",
"Definition:Propositional Function"
] | [] |
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