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-1600 | Fundamental Theorem of Algebra | Every non-constant polynomial with coefficients in $\C$ has a root in $\C$.
That is, the field of complex numbers is algebraically closed. | Let $f$ be a polynomial with coefficients in $\R$, and let $f$ have degree $n$.
By Fundamental Theorem of Arithmetic, there exists $k \in \N$ such that:
:$n = 2^k \cdot m$
with $m$ an odd number.
We proceed by induction on $k$.
Note that $f$ has roots $\alpha_1, \ldots, \alpha_n$ in some splitting field.
It suffices ... | Every non-[[Definition:Constant Polynomial|constant]] [[Definition:Polynomial|polynomial]] with [[Definition:Polynomial Coefficient|coefficients]] in $\C$ has a [[Definition:Root of Polynomial|root]] in $\C$.
That is, the [[Definition:Complex Number|field of complex numbers]] is [[Definition:Algebraically Closed Field... | Let $f$ be a polynomial with coefficients in $\R$, and let $f$ have degree $n$.
By [[Fundamental Theorem of Arithmetic]], there exists $k \in \N$ such that:
:$n = 2^k \cdot m$
with $m$ an [[Definition:Odd Integer|odd number]].
We proceed by induction on $k$.
Note that $f$ has roots $\alpha_1, \ldots, \alpha_n$ in ... | Fundamental Theorem of Algebra/Proof 4 | https://proofwiki.org/wiki/Fundamental_Theorem_of_Algebra | https://proofwiki.org/wiki/Fundamental_Theorem_of_Algebra/Proof_4 | [
"Fundamental Theorem of Algebra",
"Polynomial Theory",
"Algebra",
"Analysis",
"Fundamental Theorems"
] | [
"Definition:Constant Polynomial",
"Definition:Polynomial",
"Definition:Coefficient of Polynomial",
"Definition:Root of Polynomial",
"Definition:Complex Number",
"Definition:Algebraically Closed Field"
] | [
"Fundamental Theorem of Arithmetic",
"Definition:Odd Integer",
"Definition:Splitting Field",
"Definition:Complex Number",
"Rolle's Theorem",
"Viète's Formulas",
"Definition:Symmetric Polynomial",
"Definition:Symmetric Polynomial",
"Fundamental Theorem of Symmetric Polynomials",
"Viète's Formulas",... |
proofwiki-1601 | Fundamental Theorem of Algebra | Every non-constant polynomial with coefficients in $\C$ has a root in $\C$.
That is, the field of complex numbers is algebraically closed. | Let $\map p z$ be an arbitrary non-constant polynomial with coefficients in $\C$.
This means that for some $n \in \Z_{ge 1}$ and $a_n \ne 0$:
:$\map p z = a_n z^n + a_{n - 1}z^{n - 1} + \dots + a_0$
We have:
:$\map p z = a_n z^n \paren {1 + \dfrac {a_{n - 1} } {a_n}z^{-1} + \dots + \dfrac 1 {a_n}z^{-n} } = a_n z^n \par... | Every non-[[Definition:Constant Polynomial|constant]] [[Definition:Polynomial|polynomial]] with [[Definition:Polynomial Coefficient|coefficients]] in $\C$ has a [[Definition:Root of Polynomial|root]] in $\C$.
That is, the [[Definition:Complex Number|field of complex numbers]] is [[Definition:Algebraically Closed Field... | Let $\map p z$ be an arbitrary non-constant polynomial with coefficients in $\C$.
This means that for some $n \in \Z_{ge 1}$ and $a_n \ne 0$:
:$\map p z = a_n z^n + a_{n - 1}z^{n - 1} + \dots + a_0$
We have:
:$\map p z = a_n z^n \paren {1 + \dfrac {a_{n - 1} } {a_n}z^{-1} + \dots + \dfrac 1 {a_n}z^{-n} } = a_n z^n \p... | Fundamental Theorem of Algebra/Proof 5 | https://proofwiki.org/wiki/Fundamental_Theorem_of_Algebra | https://proofwiki.org/wiki/Fundamental_Theorem_of_Algebra/Proof_5 | [
"Fundamental Theorem of Algebra",
"Polynomial Theory",
"Algebra",
"Analysis",
"Fundamental Theorems"
] | [
"Definition:Constant Polynomial",
"Definition:Polynomial",
"Definition:Coefficient of Polynomial",
"Definition:Root of Polynomial",
"Definition:Complex Number",
"Definition:Algebraically Closed Field"
] | [] |
proofwiki-1602 | Fundamental Theorem of Algebra | Every non-constant polynomial with coefficients in $\C$ has a root in $\C$.
That is, the field of complex numbers is algebraically closed. | Let $p: \C \to \C$ be a complex, non-constant polynomial.
{{AimForCont}} that $\map p z \ne 0$ for all $z \in \C$.
Therefore by Reciprocal of Holomorphic Function $\dfrac 1 {\map p z}$ is entire.
{{begin-eqn}}
{{eqn | l = 0
| o = <
| r = \cmod {\dfrac 1 {\map p 0} }
}}
{{eqn | r = \cmod {\frac 1 {2 \pi i} \... | Every non-[[Definition:Constant Polynomial|constant]] [[Definition:Polynomial|polynomial]] with [[Definition:Polynomial Coefficient|coefficients]] in $\C$ has a [[Definition:Root of Polynomial|root]] in $\C$.
That is, the [[Definition:Complex Number|field of complex numbers]] is [[Definition:Algebraically Closed Field... | Let $p: \C \to \C$ be a [[Definition:Complex Number|complex]], non-[[Definition:Constant Polynomial|constant]] [[Definition:Polynomial|polynomial]].
{{AimForCont}} that $\map p z \ne 0$ for all $z \in \C$.
Therefore by [[Reciprocal of Holomorphic Function]] $\dfrac 1 {\map p z}$ is [[Definition:Entire Function|entire... | Fundamental Theorem of Algebra/Proof 6 | https://proofwiki.org/wiki/Fundamental_Theorem_of_Algebra | https://proofwiki.org/wiki/Fundamental_Theorem_of_Algebra/Proof_6 | [
"Fundamental Theorem of Algebra",
"Polynomial Theory",
"Algebra",
"Analysis",
"Fundamental Theorems"
] | [
"Definition:Constant Polynomial",
"Definition:Polynomial",
"Definition:Coefficient of Polynomial",
"Definition:Root of Polynomial",
"Definition:Complex Number",
"Definition:Algebraically Closed Field"
] | [
"Definition:Complex Number",
"Definition:Constant Polynomial",
"Definition:Polynomial",
"Reciprocal of Holomorphic Function",
"Definition:Entire Function",
"Cauchy's Integral Formula",
"Estimation Lemma for Contour Integrals",
"Definition:Contradiction"
] |
proofwiki-1603 | Extendability Theorem for Intersection Numbers | Let $X = \partial W$ be a smooth manifold which is the boundary of a smooth compact manifold $W$.
Let:
:$Y$ be a smooth manifold
:$Z$ be a closed smooth submanifold of $Y$
:$f: X \to Y$ be a smooth mapping.
Let there exist a smooth mapping $g: W \to Y$ such that $g \restriction_X = f$.
Then:
:$\map I {f, Z} = 0$
where ... | {{ProofWanted}}
Category:Extendability Theorem for Intersection Numbers
Category:Smooth Manifolds
Category:Named Theorems
0acfthtqqi1fi4rbdp3g4fm6k5luk72 | Let $X = \partial W$ be a [[Definition:Smooth Manifold|smooth manifold]] which is the [[Definition:Boundary (Topology)|boundary]] of a [[Definition:Smooth Manifold|smooth]] [[Definition:Compact Manifold|compact manifold]] $W$.
Let:
:$Y$ be a [[Definition:Smooth Manifold|smooth manifold]]
:$Z$ be a [[Definition:Closed ... | {{ProofWanted}}
[[Category:Extendability Theorem for Intersection Numbers]]
[[Category:Smooth Manifolds]]
[[Category:Named Theorems]]
0acfthtqqi1fi4rbdp3g4fm6k5luk72 | Extendability Theorem for Intersection Numbers | https://proofwiki.org/wiki/Extendability_Theorem_for_Intersection_Numbers | https://proofwiki.org/wiki/Extendability_Theorem_for_Intersection_Numbers | [
"Extendability Theorem for Intersection Numbers",
"Smooth Manifolds",
"Named Theorems"
] | [
"Definition:Topological Manifold/Smooth Manifold",
"Definition:Boundary (Topology)",
"Definition:Topological Manifold/Smooth Manifold",
"Definition:Compact Manifold",
"Definition:Topological Manifold/Smooth Manifold",
"Definition:Closed Manifold",
"Definition:Topological Manifold/Smooth Manifold",
"De... | [
"Category:Extendability Theorem for Intersection Numbers",
"Category:Smooth Manifolds",
"Category:Named Theorems"
] |
proofwiki-1604 | Relative Homotopy is Equivalence Relation | Let $X$ and $Y$ be topological spaces.
Let $K \subseteq X$ be a (possibly empty) subset of $X$.
Let $\map \CC {X, Y}$ be the set of all continuous mappings from $X$ to $Y$.
Define a relation $\sim$ on $\map \CC {X, Y}$ as:
:$f \sim g$ {{iff}} $f$ and $g$ are homotopic relative to $K$.
Then $\sim$ is an equivalence rela... | We examine each condition for equivalence. | Let $X$ and $Y$ be [[Definition:Topological Space|topological spaces]].
Let $K \subseteq X$ be a (possibly [[Definition:Empty Set|empty]]) [[Definition:Subset|subset]] of $X$.
Let $\map \CC {X, Y}$ be the set of all [[Definition:Continuous Mapping (Topology)|continuous mappings]] from $X$ to $Y$.
Define a [[Definiti... | We examine each condition for [[Definition:Equivalence Relation|equivalence]]. | Relative Homotopy is Equivalence Relation | https://proofwiki.org/wiki/Relative_Homotopy_is_Equivalence_Relation | https://proofwiki.org/wiki/Relative_Homotopy_is_Equivalence_Relation | [
"Homotopy Theory"
] | [
"Definition:Topological Space",
"Definition:Empty Set",
"Definition:Subset",
"Definition:Continuous Mapping (Topology)",
"Definition:Relation",
"Definition:Homotopy/Relative",
"Definition:Equivalence Relation"
] | [
"Definition:Equivalence Relation",
"Definition:Equivalence Relation"
] |
proofwiki-1605 | Homotopy Group is Homeomorphism Invariant | Let $X$ and $Y$ be two topological spaces.
Let $\phi: X \to Y$ be a homeomorphism.
Let $x_0 \in X$, $y_0 \in Y$.
Then for all $n \in \N$ the induced mapping:
:$\phi_* : \pi_n \left({X, x_0}\right) \to \pi_n \left({Y, y_0}\right):$
::$\left[{\!\left[{\, c \,}\right]\!}\right] \mapsto \left[{\!\left[{\, \phi \circ c \,}\... | Let $\phi: X \to Y$ be a homeomorphism.
We must show that:
:$(1): \quad$ If $c: \left[{0 \,.\,.\, 1}\right]^n \to X$ is a continuous mapping, then $ \phi \circ c: \left[{0 \,.\,.\, 1}\right]^n \to Y$ is also continuous
:$(2): \quad$ If $c, d: \left[{0 \,.\,.\, 1}\right]^n \to X$ are freely homotopic, then $\phi \circ c... | Let $X$ and $Y$ be two [[Definition:Topological Space|topological spaces]].
Let $\phi: X \to Y$ be a [[Definition:Homeomorphism (Topological Spaces)|homeomorphism]].
Let $x_0 \in X$, $y_0 \in Y$.
Then for all $n \in \N$ the [[Definition:Induced Mapping of Homotopy Groups|induced mapping]]:
:$\phi_* : \pi_n \left({... | Let $\phi: X \to Y$ be a [[Definition:Homeomorphism (Topological Spaces)|homeomorphism]].
We must show that:
:$(1): \quad$ If $c: \left[{0 \,.\,.\, 1}\right]^n \to X$ is a [[Definition:Everywhere Continuous Mapping (Topology)|continuous mapping]], then $ \phi \circ c: \left[{0 \,.\,.\, 1}\right]^n \to Y$ is also [[De... | Homotopy Group is Homeomorphism Invariant | https://proofwiki.org/wiki/Homotopy_Group_is_Homeomorphism_Invariant | https://proofwiki.org/wiki/Homotopy_Group_is_Homeomorphism_Invariant | [
"Homotopy Theory",
"Algebraic Topology"
] | [
"Definition:Topological Space",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Induced Mapping of Homotopy Groups",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Definition:Homotopy Group"
] | [
"Definition:Homeomorphism/Topological Spaces",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Homotopy/Free",
"Definition:Homotopy/Free",
"Definition:Homotopy/Free",
"Definition:Homotopy/Free",
"Definition:Concatenation (Topolo... |
proofwiki-1606 | Homotopy Group is Group | The set of all homotopy classes of continuous mappings:
:$c: \closedint 0 1^n \to X$
satisfying:
:$\map c {\partial \closedint 0 1^n} = x_0$
in a space $X$ at a base point $x_0$, under the operation of concatenation on class members, forms a group.
{{finish|This operation needs to be shown well-defined; probably a nast... | We examine each of the group axioms separately. | The set of all [[Definition:Homotopy Class|homotopy classes]] of [[Definition:Continuous Mapping (Topology)|continuous mappings]]:
:$c: \closedint 0 1^n \to X$
satisfying:
:$\map c {\partial \closedint 0 1^n} = x_0$
in a [[Definition:Topological Space|space]] $X$ at a base point $x_0$, under the operation of [[Definiti... | We examine each of the [[Axiom:Group Axioms|group axioms]] separately. | Homotopy Group is Group | https://proofwiki.org/wiki/Homotopy_Group_is_Group | https://proofwiki.org/wiki/Homotopy_Group_is_Group | [
"Homotopy Theory",
"Algebraic Topology",
"Examples of Groups"
] | [
"Definition:Homotopy Class",
"Definition:Continuous Mapping (Topology)",
"Definition:Topological Space",
"Definition:Concatenation (Topology)",
"Definition:Group",
"Definition:Homotopy Group"
] | [
"Axiom:Group Axioms"
] |
proofwiki-1607 | Interior Extremum Theorem | Let $f$ be a real function which is differentiable on the open interval $\openint a b$.
Let $f$ have a local minimum or local maximum at $\xi \in \openint a b$.
Then:
:$\map {f'} \xi = 0$ | By definition of derivative at a point:
:$\dfrac {\map f x - \map f \xi} {x - \xi} \to \map {f'} \xi$ as $x \to \xi$
Suppose $\map {f'} \xi > 0$.
Then from Behaviour of Function Near Limit it follows that:
:$\exists I = \openint {\xi - h} {\xi + h}: \dfrac {\map f x - \map f \xi} {x - \xi} > 0$
provided that $x \in I$ ... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Differentiable on Interval|differentiable]] on the [[Definition:Open Real Interval|open interval]] $\openint a b$.
Let $f$ have a [[Definition:Local Minimum|local minimum]] or [[Definition:Local Maximum|local maximum]] at $\xi \in \openint a... | By definition of [[Definition:Derivative of Real Function at Point|derivative at a point]]:
:$\dfrac {\map f x - \map f \xi} {x - \xi} \to \map {f'} \xi$ as $x \to \xi$
Suppose $\map {f'} \xi > 0$.
Then from [[Behaviour of Function Near Limit]] it follows that:
:$\exists I = \openint {\xi - h} {\xi + h}: \dfrac {\m... | Interior Extremum Theorem/Proof 1 | https://proofwiki.org/wiki/Interior_Extremum_Theorem | https://proofwiki.org/wiki/Interior_Extremum_Theorem/Proof_1 | [
"Interior Extremum Theorem",
"Local Maxima",
"Local Minima",
"Derivative Tests",
"Differential Calculus",
"Named Theorems"
] | [
"Definition:Real Function",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Real Interval/Open",
"Definition:Minimum Value of Real Function/Local",
"Definition:Maximum Value of Real Function/Local"
] | [
"Definition:Derivative/Real Function/Derivative at Point",
"Behaviour of Function Near Limit",
"Definition:Real Interval/Open",
"Definition:Minimum Value of Real Function/Local",
"Definition:Real Interval/Open",
"Definition:Maximum Value of Real Function/Local"
] |
proofwiki-1608 | Interior Extremum Theorem | Let $f$ be a real function which is differentiable on the open interval $\openint a b$.
Let $f$ have a local minimum or local maximum at $\xi \in \openint a b$.
Then:
:$\map {f'} \xi = 0$ | By definition of local maximum:
:$(1): \quad \map f {\xi + \epsilon} - \map f \xi < 0$
for sufficiently small $\epsilon \in \R_{>0}$.
Similarly, definition of local minimum:
:$(2): \quad \map f {\xi + \epsilon} - \map f \xi > 0$
for sufficiently small $\epsilon \in \R_{>0}$.
Let it be assumed that $\map f {\xi + \epsil... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Differentiable on Interval|differentiable]] on the [[Definition:Open Real Interval|open interval]] $\openint a b$.
Let $f$ have a [[Definition:Local Minimum|local minimum]] or [[Definition:Local Maximum|local maximum]] at $\xi \in \openint a... | By definition of [[Definition:Local Maximum|local maximum]]:
:$(1): \quad \map f {\xi + \epsilon} - \map f \xi < 0$
for [[Definition:Sufficiently Small|sufficiently small]] $\epsilon \in \R_{>0}$.
Similarly, definition of [[Definition:Local Minimum|local minimum]]:
:$(2): \quad \map f {\xi + \epsilon} - \map f \xi > 0... | Interior Extremum Theorem/Proof 2 | https://proofwiki.org/wiki/Interior_Extremum_Theorem | https://proofwiki.org/wiki/Interior_Extremum_Theorem/Proof_2 | [
"Interior Extremum Theorem",
"Local Maxima",
"Local Minima",
"Derivative Tests",
"Differential Calculus",
"Named Theorems"
] | [
"Definition:Real Function",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Real Interval/Open",
"Definition:Minimum Value of Real Function/Local",
"Definition:Maximum Value of Real Function/Local"
] | [
"Definition:Maximum Value of Real Function/Local",
"Definition:Sufficiently Small",
"Definition:Minimum Value of Real Function/Local",
"Definition:Sufficiently Small",
"Taylor's Theorem",
"Definition:Positive/Integer",
"Definition:Power (Algebra)/Integer",
"Definition:Big-O Notation",
"Definition:Bo... |
proofwiki-1609 | Behaviour of Function Near Limit | Let $f$ be a real function.
Let $f \left({x}\right) \to l$ as $x \to \xi$.
Then:
:If $l > 0$, then $\exists h > 0: \forall x: \xi - h < x < \xi + h, x \ne \xi: \map f x > 0$
:If $l < 0$, then $\exists h > 0: \forall x: \xi - h < x < \xi + h, x \ne \xi: \map f x < 0$ | From the definition of limit of a function:
:$\forall \epsilon > 0: \exists \delta > 0: 0 < \size {x - \xi} < \delta \implies \size {\map f x - l} < \epsilon$
Let $l > 0$.
Since this is true for all $\epsilon > 0$, it is also true for $\epsilon = l$.
So let the value of $\delta$, for the above to be true, labelled $h$.... | Let $f$ be a [[Definition:Real Function|real function]].
Let $f \left({x}\right) \to l$ as $x \to \xi$.
Then:
:If $l > 0$, then $\exists h > 0: \forall x: \xi - h < x < \xi + h, x \ne \xi: \map f x > 0$
:If $l < 0$, then $\exists h > 0: \forall x: \xi - h < x < \xi + h, x \ne \xi: \map f x < 0$ | From the definition of [[Definition:Limit of Real Function|limit of a function]]:
:$\forall \epsilon > 0: \exists \delta > 0: 0 < \size {x - \xi} < \delta \implies \size {\map f x - l} < \epsilon$
Let $l > 0$.
Since this is true for all $\epsilon > 0$, it is also true for $\epsilon = l$.
So let the value of $\delt... | Behaviour of Function Near Limit | https://proofwiki.org/wiki/Behaviour_of_Function_Near_Limit | https://proofwiki.org/wiki/Behaviour_of_Function_Near_Limit | [
"Limits of Real Functions"
] | [
"Definition:Real Function"
] | [
"Definition:Limit of Real Function"
] |
proofwiki-1610 | Fundamental Group is Independent of Base Point for Path-Connected Space | Let $X$ be a path-connected space.
For $x \in X$, let $\map {\pi_1}{X, x}$ denote the fundamental group.
For $x, y \in X$, there is an isomorphism:
:$\phi: \map {\pi_1} {X, x} \to \map {\pi_1} {X, y}$ | Since $X$ is path-connected there exists a path $f$ connecting $y$ and $x$.
We define $\phi_f: \map {\pi_1} {X, x} \to \map {\pi_1} {X, y}$ by $\map {\phi_f} {\sqbrk g} =\sqbrk {f^{-1} g f}$.
$\phi$ is a homomorphism of groups, as is seen from:
:$\ds \map {\phi_f} {\sqbrk {g h} } = \sqbrk {f^{-1} g h f} = \sqbrk {f^{-1... | Let $X$ be a [[Definition:Path-Connected|path-connected]] [[Definition:Topological Space|space]].
For $x \in X$, let $\map {\pi_1}{X, x}$ denote the [[Definition:Fundamental Group|fundamental group]].
For $x, y \in X$, there is an [[Definition:Group Isomorphism|isomorphism]]:
:$\phi: \map {\pi_1} {X, x} \to \map {\p... | Since $X$ is [[Definition:Path-Connected|path-connected]] there exists a [[Definition:Path (Topology)|path]] $f$ connecting $y$ and $x$.
We define $\phi_f: \map {\pi_1} {X, x} \to \map {\pi_1} {X, y}$ by $\map {\phi_f} {\sqbrk g} =\sqbrk {f^{-1} g f}$.
$\phi$ is a [[Definition:Group Homomorphism|homomorphism of group... | Fundamental Group is Independent of Base Point for Path-Connected Space | https://proofwiki.org/wiki/Fundamental_Group_is_Independent_of_Base_Point_for_Path-Connected_Space | https://proofwiki.org/wiki/Fundamental_Group_is_Independent_of_Base_Point_for_Path-Connected_Space | [
"Algebraic Topology"
] | [
"Definition:Path-Connected",
"Definition:Topological Space",
"Definition:Fundamental Group",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism"
] | [
"Definition:Path-Connected",
"Definition:Path (Topology)",
"Definition:Group Homomorphism",
"Definition:Group Homomorphism",
"Definition:Bijection",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Category:Algebraic Topology"
] |
proofwiki-1611 | List of Fundamental Groups for 2-Manifolds | {{MissingLinks|two-mainfold, also review links for everything else because it's all over the place}}
For the following two-manifolds, the fundamental group for any point in $X$, written $\map {\pi_1} X$ is isomorphic to the listed group:
:$\map {\pi_1} {\Bbb S^1 \times \closedint 0 1} = \Z$
:$\map {\pi_1} {\Bbb S^2} = ... | {{ProofWanted}}
Category:Algebraic Topology
0dl7p0jv58blp4y4j335ztzml678qi2 | {{MissingLinks|two-mainfold, also review links for everything else because it's all over the place}}
For the following two-manifolds, the [[Definition:Homotopy Group|fundamental group]] for [[Fundamental Group is Independent of Base Point for Path-Connected Space|any point in $X$]], written $\map {\pi_1} X$ is [[Defin... | {{ProofWanted}}
[[Category:Algebraic Topology]]
0dl7p0jv58blp4y4j335ztzml678qi2 | List of Fundamental Groups for 2-Manifolds | https://proofwiki.org/wiki/List_of_Fundamental_Groups_for_2-Manifolds | https://proofwiki.org/wiki/List_of_Fundamental_Groups_for_2-Manifolds | [
"Algebraic Topology"
] | [
"Definition:Homotopy Group",
"Fundamental Group is Independent of Base Point for Path-Connected Space",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Definition:Trivial Group"
] | [
"Category:Algebraic Topology"
] |
proofwiki-1612 | Rolle's Theorem | Let $f$ be a real function which is:
:continuous on the closed interval $\closedint a b$
and:
:differentiable on the open interval $\openint a b$.
Let $\map f a = \map f b$.
Then:
:$\exists \xi \in \openint a b: \map {f'} \xi = 0$ | We have that $f$ is continuous on $\closedint a b$.
It follows from Continuous Image of Closed Real Interval is Closed Real Interval that $f$ attains:
:a maximum $M$ at some $\xi_1 \in \closedint a b$
and:
:a minimum $m$ at some $\xi_2 \in \closedint a b$.
Suppose $\xi_1$ and $\xi_2$ are both end points of $\closedint ... | Let $f$ be a [[Definition:Real Function|real function]] which is:
:[[Definition:Continuous on Interval|continuous]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$
and:
:[[Definition:Differentiable on Interval|differentiable]] on the [[Definition:Open Real Interval|open interval]] $\openint ... | We have that $f$ is [[Definition:Continuous on Interval|continuous]] on $\closedint a b$.
It follows from [[Continuous Image of Closed Real Interval is Closed Real Interval]] that $f$ attains:
:a [[Definition:Maximum Value|maximum]] $M$ at some $\xi_1 \in \closedint a b$
and:
:a [[Definition:Minimum Value|minimum]] $m... | Rolle's Theorem/Proof 1 | https://proofwiki.org/wiki/Rolle's_Theorem | https://proofwiki.org/wiki/Rolle's_Theorem/Proof_1 | [
"Rolle's Theorem",
"Differentiable Real Functions",
"Continuous Real Functions"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Interval",
"Definition:Real Interval/Closed",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Real Interval/Open"
] | [
"Definition:Continuous Real Function/Interval",
"Continuous Image of Closed Real Interval is Closed Real Interval",
"Definition:Maximum Value of Real Function/Absolute",
"Definition:Minimum Value of Real Function/Absolute",
"Definition:Constant Mapping",
"Derivative of Constant",
"Definition:Maximum Val... |
proofwiki-1613 | Rolle's Theorem | Let $f$ be a real function which is:
:continuous on the closed interval $\closedint a b$
and:
:differentiable on the open interval $\openint a b$.
Let $\map f a = \map f b$.
Then:
:$\exists \xi \in \openint a b: \map {f'} \xi = 0$ | First take the case where:
:$\forall x \in \openint a b: \map f x = 0$
Then:
:$\forall x \in \openint a b: \map {f'} x = 0$
Otherwise:
:$\exists c \in \openint a b: \map f c \ne 0$
Let $\map f c > 0$.
Then there exists an absolute maximum at a point $\xi \in \openint a b$.
Hence:
{{begin-eqn}}
{{eqn | l = \dfrac {\map ... | Let $f$ be a [[Definition:Real Function|real function]] which is:
:[[Definition:Continuous on Interval|continuous]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$
and:
:[[Definition:Differentiable on Interval|differentiable]] on the [[Definition:Open Real Interval|open interval]] $\openint ... | First take the case where:
:$\forall x \in \openint a b: \map f x = 0$
Then:
:$\forall x \in \openint a b: \map {f'} x = 0$
Otherwise:
:$\exists c \in \openint a b: \map f c \ne 0$
Let $\map f c > 0$.
Then there exists an [[Definition:Absolute Maximum|absolute maximum]] at a point $\xi \in \openint a b$.
Hence:
... | Rolle's Theorem/Proof 2 | https://proofwiki.org/wiki/Rolle's_Theorem | https://proofwiki.org/wiki/Rolle's_Theorem/Proof_2 | [
"Rolle's Theorem",
"Differentiable Real Functions",
"Continuous Real Functions"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Interval",
"Definition:Real Interval/Closed",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Real Interval/Open"
] | [
"Definition:Maximum Value of Real Function/Absolute",
"Definition:Positive/Real Number",
"Definition:Negative/Real Number",
"Definition:Minimum Value of Real Function/Absolute",
"Definition:Positive/Real Number",
"Definition:Negative/Real Number"
] |
proofwiki-1614 | Rolle's Theorem | Let $f$ be a real function which is:
:continuous on the closed interval $\closedint a b$
and:
:differentiable on the open interval $\openint a b$.
Let $\map f a = \map f b$.
Then:
:$\exists \xi \in \openint a b: \map {f'} \xi = 0$ | Let the function $g$ be defined as:
:$\map g t = \map {R_n} t - \dfrac {\paren {t - a}^{n + 1} } {\paren {x - a}^{n + 1} } \map {R_n} x$
Then:
:$\map {g^{\paren k} } a = 0$
for $k = 0, \dotsc, n$, and $\map g x = 0$.
Apply Rolle's Theorem successively to $g, g', \dotsc, g^{\paren n}$.
Then there exist:
:$\xi_1, \ldots,... | Let $f$ be a [[Definition:Real Function|real function]] which is:
:[[Definition:Continuous on Interval|continuous]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$
and:
:[[Definition:Differentiable on Interval|differentiable]] on the [[Definition:Open Real Interval|open interval]] $\openint ... | Let the [[Definition:Real Function|function]] $g$ be defined as:
:$\map g t = \map {R_n} t - \dfrac {\paren {t - a}^{n + 1} } {\paren {x - a}^{n + 1} } \map {R_n} x$
Then:
:$\map {g^{\paren k} } a = 0$
for $k = 0, \dotsc, n$, and $\map g x = 0$.
Apply [[Rolle's Theorem]] successively to $g, g', \dotsc, g^{\paren n}$... | Taylor's Theorem/One Variable/Proof by Rolle's Theorem | https://proofwiki.org/wiki/Rolle's_Theorem | https://proofwiki.org/wiki/Taylor's_Theorem/One_Variable/Proof_by_Rolle's_Theorem | [
"Rolle's Theorem",
"Differentiable Real Functions",
"Continuous Real Functions"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Interval",
"Definition:Real Interval/Closed",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Real Interval/Open"
] | [
"Definition:Real Function",
"Rolle's Theorem"
] |
proofwiki-1615 | Mean Value Theorem | Let $f$ be a real function which is continuous on the closed interval $\closedint a b$ and differentiable on the open interval $\openint a b$.
Then:
:$\exists \xi \in \openint a b: \map {f'} \xi = \dfrac {\map f b - \map f a} {b - a}$ | From Continuous Real Function is Darboux Integrable, $f$ is Darboux integrable on $\closedint a b$.
By the Extreme Value Theorem, there exist $m, M \in \closedint a b$ such that:
:$\ds \map f m = \min_{x \mathop \in \closedint a b} \map f x$
:$\ds \map f M = \max_{x \mathop \in \closedint a b} \map f x$
Then, from Darb... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Continuous Real Function on Closed Interval|continuous]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$ and [[Definition:Differentiable on Interval|differentiable]] on the [[Definition:Open Real Interval|open inte... | From [[Continuous Real Function is Darboux Integrable]], $f$ is [[Definition:Darboux Integrable Function|Darboux integrable]] on $\closedint a b$.
By the [[Extreme Value Theorem]], there exist $m, M \in \closedint a b$ such that:
:$\ds \map f m = \min_{x \mathop \in \closedint a b} \map f x$
:$\ds \map f M = \max_{x ... | Mean Value Theorem for Integrals/Proof 1 | https://proofwiki.org/wiki/Mean_Value_Theorem | https://proofwiki.org/wiki/Mean_Value_Theorem_for_Integrals/Proof_1 | [
"Mean Value Theorem",
"Differential Calculus",
"Named Theorems"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Closed Interval",
"Definition:Real Interval/Closed",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Real Interval/Open"
] | [
"Continuous Real Function is Darboux Integrable",
"Definition:Darboux Integrable Function",
"Extreme Value Theorem",
"Darboux's Theorem",
"Intermediate Value Theorem"
] |
proofwiki-1616 | Mean Value Theorem | Let $f$ be a real function which is continuous on the closed interval $\closedint a b$ and differentiable on the open interval $\openint a b$.
Then:
:$\exists \xi \in \openint a b: \map {f'} \xi = \dfrac {\map f b - \map f a} {b - a}$ | From Continuous Real Function is Darboux Integrable, $f$ is Darboux integrable on $\closedint a b$.
Let $F : \closedint a b \to \R$ be a real function defined by:
:$\ds \map F x = \int_a^x \map f x \rd x$
We are assured that this function is well-defined, since $f$ is integrable on $\closedint a b$.
From Fundamental T... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Continuous Real Function on Closed Interval|continuous]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$ and [[Definition:Differentiable on Interval|differentiable]] on the [[Definition:Open Real Interval|open inte... | From [[Continuous Real Function is Darboux Integrable]], $f$ is [[Definition:Darboux Integrable Function|Darboux integrable]] on $\closedint a b$.
Let $F : \closedint a b \to \R$ be a [[Definition:Real Function|real function]] defined by:
:$\ds \map F x = \int_a^x \map f x \rd x$
We are assured that this function i... | Mean Value Theorem for Integrals/Proof 2 | https://proofwiki.org/wiki/Mean_Value_Theorem | https://proofwiki.org/wiki/Mean_Value_Theorem_for_Integrals/Proof_2 | [
"Mean Value Theorem",
"Differential Calculus",
"Named Theorems"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Closed Interval",
"Definition:Real Interval/Closed",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Real Interval/Open"
] | [
"Continuous Real Function is Darboux Integrable",
"Definition:Darboux Integrable Function",
"Definition:Real Function",
"Definition:Well-Defined",
"Definition:Darboux Integrable Function",
"Fundamental Theorem of Calculus/First Part",
"Definition:Continuous Real Function",
"Definition:Differentiable M... |
proofwiki-1617 | Mean Value Theorem | Let $f$ be a real function which is continuous on the closed interval $\closedint a b$ and differentiable on the open interval $\openint a b$.
Then:
:$\exists \xi \in \openint a b: \map {f'} \xi = \dfrac {\map f b - \map f a} {b - a}$ | For any constant $h \in \R$ we may construct the real function defined on $\closedint a b$ by:
:$\map F x = \map f x + h x$
We have that $h x$ is continuous on $\closedint a b$ from Linear Function is Continuous.
From the Sum Rule for Continuous Real Functions, $F$ is continuous on $\closedint a b$ and differentiable o... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Continuous Real Function on Closed Interval|continuous]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$ and [[Definition:Differentiable on Interval|differentiable]] on the [[Definition:Open Real Interval|open inte... | For any [[Definition:Constant|constant]] $h \in \R$ we may construct the [[Definition:Real Function|real function]] defined on $\closedint a b$ by:
:$\map F x = \map f x + h x$
We have that $h x$ is [[Definition:Continuous on Interval|continuous]] on $\closedint a b$ from [[Linear Function is Continuous]].
From the [... | Mean Value Theorem/Proof 1 | https://proofwiki.org/wiki/Mean_Value_Theorem | https://proofwiki.org/wiki/Mean_Value_Theorem/Proof_1 | [
"Mean Value Theorem",
"Differential Calculus",
"Named Theorems"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Closed Interval",
"Definition:Real Interval/Closed",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Real Interval/Open"
] | [
"Definition:Constant",
"Definition:Real Function",
"Definition:Continuous Real Function/Interval",
"Linear Function is Continuous",
"Combination Theorem for Continuous Functions/Real/Sum Rule",
"Definition:Continuous Real Function/Closed Interval",
"Definition:Differentiable Mapping/Real Function/Interv... |
proofwiki-1618 | Mean Value Theorem | Let $f$ be a real function which is continuous on the closed interval $\closedint a b$ and differentiable on the open interval $\openint a b$.
Then:
:$\exists \xi \in \openint a b: \map {f'} \xi = \dfrac {\map f b - \map f a} {b - a}$ | Let $g : \closedint a b \to \R$ be a real function with:
:$\map g x = x$
for all $x \in \closedint a b$.
By Power Rule for Derivatives, we have:
:$g$ is differentiable with $\map {g'} x = 1$ for all $x \in \closedint a b$.
Note that in particular:
:$\map {g'} x \ne 0$ for all $x \in \openint a b$.
Since $f$ is cont... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Continuous Real Function on Closed Interval|continuous]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$ and [[Definition:Differentiable on Interval|differentiable]] on the [[Definition:Open Real Interval|open inte... | Let $g : \closedint a b \to \R$ be a [[Definition:Real Function|real function]] with:
:$\map g x = x$
for all $x \in \closedint a b$.
By [[Power Rule for Derivatives]], we have:
:$g$ is [[Definition:Differentiable Real Function|differentiable]] with $\map {g'} x = 1$ for all $x \in \closedint a b$.
Note that in... | Mean Value Theorem/Proof 2 | https://proofwiki.org/wiki/Mean_Value_Theorem | https://proofwiki.org/wiki/Mean_Value_Theorem/Proof_2 | [
"Mean Value Theorem",
"Differential Calculus",
"Named Theorems"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Closed Interval",
"Definition:Real Interval/Closed",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Real Interval/Open"
] | [
"Definition:Real Function",
"Power Rule for Derivatives",
"Definition:Differentiable Mapping/Real Function",
"Definition:Continuous Real Function/Closed Interval",
"Definition:Differentiable Mapping/Real Function/Interval",
"Cauchy Mean Value Theorem"
] |
proofwiki-1619 | Mean Value Theorem | Let $f$ be a real function which is continuous on the closed interval $\closedint a b$ and differentiable on the open interval $\openint a b$.
Then:
:$\exists \xi \in \openint a b: \map {f'} \xi = \dfrac {\map f b - \map f a} {b - a}$ | Recall the Cauchy Mean Value Theorem:
{{:Cauchy Mean Value Theorem}}
The result follows by setting $\map g x = x$ for all $x \in \R$.
{{qed}} | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Continuous Real Function on Closed Interval|continuous]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$ and [[Definition:Differentiable on Interval|differentiable]] on the [[Definition:Open Real Interval|open inte... | Recall the [[Cauchy Mean Value Theorem]]:
{{:Cauchy Mean Value Theorem}}
The result follows by setting $\map g x = x$ for all $x \in \R$.
{{qed}} | Mean Value Theorem/Proof 3 | https://proofwiki.org/wiki/Mean_Value_Theorem | https://proofwiki.org/wiki/Mean_Value_Theorem/Proof_3 | [
"Mean Value Theorem",
"Differential Calculus",
"Named Theorems"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Closed Interval",
"Definition:Real Interval/Closed",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Real Interval/Open"
] | [
"Cauchy Mean Value Theorem"
] |
proofwiki-1620 | Mean Value Theorem | Let $f$ be a real function which is continuous on the closed interval $\closedint a b$ and differentiable on the open interval $\openint a b$.
Then:
:$\exists \xi \in \openint a b: \map {f'} \xi = \dfrac {\map f b - \map f a} {b - a}$ | Let $G$ be a real-valued function continuous on $\closedint a x$ and differentiable with non-vanishing derivative on $\openint a x$.
Let:
:$\map F t = \map f t + \dfrac {\map {f'} t} {1!} \paren {x - t} + \dotsb + \dfrac {\map {f^{\paren n} } t} {n!} \paren {x - t}^n$
By the Cauchy Mean Value Theorem:
:$(1): \quad \dfr... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Continuous Real Function on Closed Interval|continuous]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$ and [[Definition:Differentiable on Interval|differentiable]] on the [[Definition:Open Real Interval|open inte... | Let $G$ be a [[Definition:Real-Valued Function|real-valued function]] [[Definition:Continuous Real Function|continuous]] on $\closedint a x$ and [[Definition:Differentiable Real Function|differentiable]] with non-vanishing [[Definition:Derivative|derivative]] on $\openint a x$.
Let:
:$\map F t = \map f t + \dfrac {\ma... | Taylor's Theorem/One Variable/Proof by Cauchy Mean Value Theorem | https://proofwiki.org/wiki/Mean_Value_Theorem | https://proofwiki.org/wiki/Taylor's_Theorem/One_Variable/Proof_by_Cauchy_Mean_Value_Theorem | [
"Mean Value Theorem",
"Differential Calculus",
"Named Theorems"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Closed Interval",
"Definition:Real Interval/Closed",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Real Interval/Open"
] | [
"Definition:Real-Valued Function",
"Definition:Continuous Real Function",
"Definition:Differentiable Mapping/Real Function",
"Definition:Derivative",
"Cauchy Mean Value Theorem",
"Definition:Fraction/Numerator",
"Definition:Taylor Series/Remainder/Lagrange Form",
"Definition:Taylor Series/Remainder/Ca... |
proofwiki-1621 | Zero Derivative implies Constant Function | Let $f$ be a real function which is continuous on the closed interval $\closedint a b$ and differentiable on the open interval $\openint a b$.
Suppose that:
:$\forall x \in \openint a b: \map {f'} x = 0$
Then $f$ is constant on $\closedint a b$. | When $x = a$ then $\map f x = \map f a$ by definition of mapping.
Otherwise, let $x \in \hointl a b$.
We have that:
:$f$ is continuous on the closed interval $\closedint a b$
:$f$ is differentiable on the open interval $\openint a b$
Hence it satisfies the conditions of the Mean Value Theorem on $\closedint a b$.
Hence... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Continuous on Interval|continuous]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$ and [[Definition:Differentiable on Interval|differentiable]] on the [[Definition:Open Real Interval|open interval]] $\openint a b$... | When $x = a$ then $\map f x = \map f a$ by definition of [[Definition:Mapping|mapping]].
Otherwise, let $x \in \hointl a b$.
We have that:
:$f$ is [[Definition:Continuous on Interval|continuous]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$
:$f$ is [[Definition:Differentiable on Interv... | Zero Derivative implies Constant Function | https://proofwiki.org/wiki/Zero_Derivative_implies_Constant_Function | https://proofwiki.org/wiki/Zero_Derivative_implies_Constant_Function | [
"Differential Calculus",
"Constant Mappings"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Interval",
"Definition:Real Interval/Closed",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Real Interval/Open",
"Definition:Constant Mapping"
] | [
"Definition:Mapping",
"Definition:Continuous Real Function/Interval",
"Definition:Real Interval/Closed",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Real Interval/Open",
"Mean Value Theorem"
] |
proofwiki-1622 | Ostrowski's Theorem | Every non-trivial norm on the rational numbers $\Q$ is equivalent to either:
:the $p$-adic norm $\norm {\, \cdot \,}_p$ for some prime $p$
or:
:the absolute value, $\size {\, \cdot \,}$. | Let $\norm {\, \cdot \,}$ be a non-trivial norm on the rational numbers $\Q$. | Every [[Definition:Nontrivial Division Ring Norm|non-trivial]] [[Definition:Norm on Division Ring|norm]] on the [[Definition:Rational Numbers|rational numbers]] $\Q$ is [[Definition:Equivalent Division Ring Norms|equivalent]] to either:
:the [[Definition:P-adic Norm|$p$-adic norm $\norm {\, \cdot \,}_p$]] for some [[De... | Let $\norm {\, \cdot \,}$ be a [[Definition:Nontrivial Division Ring Norm|non-trivial]] [[Definition:Norm on Division Ring|norm]] on the [[Definition:Rational Numbers|rational numbers]] $\Q$. | Ostrowski's Theorem | https://proofwiki.org/wiki/Ostrowski's_Theorem | https://proofwiki.org/wiki/Ostrowski's_Theorem | [
"Normed Division Rings",
"P-adic Number Theory",
"Ostrowski's Theorem"
] | [
"Definition:Trivial Norm/Division Ring/Nontrivial",
"Definition:Norm/Division Ring",
"Definition:Rational Number",
"Definition:Equivalent Division Ring Norms",
"Definition:P-adic Norm",
"Definition:Prime Number",
"Definition:Absolute Value"
] | [
"Definition:Trivial Norm/Division Ring/Nontrivial",
"Definition:Norm/Division Ring",
"Definition:Rational Number"
] |
proofwiki-1623 | Cauchy Mean Value Theorem | Let $f$ and $g$ be real functions which are continuous on the closed interval $\closedint a b$ and differentiable on the open interval $\openint a b$.
Suppose:
:$\forall x \in \openint a b: \map {g'} x \ne 0$
Then:
:$\exists \xi \in \openint a b: \dfrac {\map {f'} \xi} {\map {g'} \xi} = \dfrac {\map f b - \map f a} {\m... | First we check $\map g a \ne \map g b$.
{{AimForCont}} $\map g a = \map g b$.
From Rolle's Theorem:
:$\exists \xi \in \openint a b: \map {g'} \xi = 0$.
This contradicts $\forall x \in \openint a b: \map {g'} x \ne 0$.
Thus by Proof by Contradiction $\map g a \ne \map g b$.
Let $h = \dfrac {\map f b - \map f a} {\map g ... | Let $f$ and $g$ be [[Definition:Real Function|real functions]] which are [[Definition:Continuous on Interval|continuous]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$ and [[Definition:Differentiable on Interval|differentiable]] on the [[Definition:Open Real Interval|open interval]] $\open... | First we check $\map g a \ne \map g b$.
{{AimForCont}} $\map g a = \map g b$.
From [[Rolle's Theorem]]:
:$\exists \xi \in \openint a b: \map {g'} \xi = 0$.
This contradicts $\forall x \in \openint a b: \map {g'} x \ne 0$.
Thus by [[Proof by Contradiction]] $\map g a \ne \map g b$.
Let $h = \dfrac {\map f b - \ma... | Cauchy Mean Value Theorem | https://proofwiki.org/wiki/Cauchy_Mean_Value_Theorem | https://proofwiki.org/wiki/Cauchy_Mean_Value_Theorem | [
"Cauchy Mean Value Theorem",
"Differential Calculus"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Interval",
"Definition:Real Interval/Closed",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Real Interval/Open"
] | [
"Rolle's Theorem",
"Proof by Contradiction",
"Definition:Real Function",
"Sum Rule for Derivatives",
"Derivative of Constant Multiple",
"Rolle's Theorem"
] |
proofwiki-1624 | Cauchy Mean Value Theorem | Let $f$ and $g$ be real functions which are continuous on the closed interval $\closedint a b$ and differentiable on the open interval $\openint a b$.
Suppose:
:$\forall x \in \openint a b: \map {g'} x \ne 0$
Then:
:$\exists \xi \in \openint a b: \dfrac {\map {f'} \xi} {\map {g'} \xi} = \dfrac {\map f b - \map f a} {\m... | Let $G$ be a real-valued function continuous on $\closedint a x$ and differentiable with non-vanishing derivative on $\openint a x$.
Let:
:$\map F t = \map f t + \dfrac {\map {f'} t} {1!} \paren {x - t} + \dotsb + \dfrac {\map {f^{\paren n} } t} {n!} \paren {x - t}^n$
By the Cauchy Mean Value Theorem:
:$(1): \quad \dfr... | Let $f$ and $g$ be [[Definition:Real Function|real functions]] which are [[Definition:Continuous on Interval|continuous]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$ and [[Definition:Differentiable on Interval|differentiable]] on the [[Definition:Open Real Interval|open interval]] $\open... | Let $G$ be a [[Definition:Real-Valued Function|real-valued function]] [[Definition:Continuous Real Function|continuous]] on $\closedint a x$ and [[Definition:Differentiable Real Function|differentiable]] with non-vanishing [[Definition:Derivative|derivative]] on $\openint a x$.
Let:
:$\map F t = \map f t + \dfrac {\ma... | Taylor's Theorem/One Variable/Proof by Cauchy Mean Value Theorem | https://proofwiki.org/wiki/Cauchy_Mean_Value_Theorem | https://proofwiki.org/wiki/Taylor's_Theorem/One_Variable/Proof_by_Cauchy_Mean_Value_Theorem | [
"Cauchy Mean Value Theorem",
"Differential Calculus"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Interval",
"Definition:Real Interval/Closed",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Real Interval/Open"
] | [
"Definition:Real-Valued Function",
"Definition:Continuous Real Function",
"Definition:Differentiable Mapping/Real Function",
"Definition:Derivative",
"Cauchy Mean Value Theorem",
"Definition:Fraction/Numerator",
"Definition:Taylor Series/Remainder/Lagrange Form",
"Definition:Taylor Series/Remainder/Ca... |
proofwiki-1625 | Homology Group is Group | The $p$th singular homology group of a space $X$ is a group. | {{ProofWanted}}
Category:Homology Groups
mxxmlkcw505h513v0osmyasyxkzb38y | The [[Definition:Homology Group|$p$th singular homology group]] of a space $X$ is a group. | {{ProofWanted}}
[[Category:Homology Groups]]
mxxmlkcw505h513v0osmyasyxkzb38y | Homology Group is Group | https://proofwiki.org/wiki/Homology_Group_is_Group | https://proofwiki.org/wiki/Homology_Group_is_Group | [
"Homology Groups"
] | [
"Definition:Homology Group"
] | [
"Category:Homology Groups"
] |
proofwiki-1626 | Limit of Absolute Value | Let $x, \xi \in \R$ be real numbers.
Then:
:$\size {x - \xi} \to 0$ as $x \to \xi$
where $\size {x - \xi}$ denotes the Absolute Value. | Let $\epsilon > 0$.
Let $\delta = \epsilon$.
From the definition of a limit of a function, we need to show that $\size {\map f x - 0} < \epsilon$ provided that $0 < \size {x - \xi} < \delta$, where $\map f x = \size {x - \xi}$.
Thus, provided $0 < \size {x - \xi} < \delta$, we have:
{{begin-eqn}}
{{eqn | l = \size {x -... | Let $x, \xi \in \R$ be [[Definition:Real Number|real numbers]].
Then:
:$\size {x - \xi} \to 0$ as $x \to \xi$
where $\size {x - \xi}$ denotes the [[Definition:Absolute Value|Absolute Value]]. | Let $\epsilon > 0$.
Let $\delta = \epsilon$.
From the definition of a [[Definition:Limit of Real Function|limit of a function]], we need to show that $\size {\map f x - 0} < \epsilon$ provided that $0 < \size {x - \xi} < \delta$, where $\map f x = \size {x - \xi}$.
Thus, provided $0 < \size {x - \xi} < \delta$, we ... | Limit of Absolute Value | https://proofwiki.org/wiki/Limit_of_Absolute_Value | https://proofwiki.org/wiki/Limit_of_Absolute_Value | [
"Examples of Limits of Real Functions",
"Absolute Value Function"
] | [
"Definition:Real Number",
"Definition:Absolute Value"
] | [
"Definition:Limit of Real Function"
] |
proofwiki-1627 | Limit of Function in Interval | Let $f$ be a real function which is defined on the open interval $\openint a b$.
Let $\xi \in \openint a b$
Suppose that, $\forall x \in \openint a b$, either:
:$\xi \le \map f x \le x$
or:
:$x \le \map f x \le \xi$
Then $\map f x \to \xi$ as $x \to \xi$. | Note that $\size {\map f x - \xi} \le \size {\xi - x}$.
From Limit of Absolute Value we have that $\size {x - \xi} \to 0$ as $x \to \xi$.
The result follows from the Squeeze Theorem.
{{qed}} | Let $f$ be a [[Definition:Real Function|real function]] which is defined on the [[Definition:Open Real Interval|open interval]] $\openint a b$.
Let $\xi \in \openint a b$
Suppose that, $\forall x \in \openint a b$, either:
:$\xi \le \map f x \le x$
or:
:$x \le \map f x \le \xi$
Then $\map f x \to \xi$ as $x \to \xi... | Note that $\size {\map f x - \xi} \le \size {\xi - x}$.
From [[Limit of Absolute Value]] we have that $\size {x - \xi} \to 0$ as $x \to \xi$.
The result follows from the [[Squeeze Theorem]].
{{qed}} | Limit of Function in Interval | https://proofwiki.org/wiki/Limit_of_Function_in_Interval | https://proofwiki.org/wiki/Limit_of_Function_in_Interval | [
"Limits of Real Functions"
] | [
"Definition:Real Function",
"Definition:Real Interval/Open"
] | [
"Limit of Absolute Value",
"Squeeze Theorem"
] |
proofwiki-1628 | Strictly Monotone Real Function is Bijective | Let $f$ be a real function which is defined on $I \subseteq \R$.
Let $f$ be strictly monotone on $I$.
Let the image of $f$ be $J$.
Then $f: I \to J$ is a bijection. | From Strictly Monotone Mapping with Totally Ordered Domain is Injective, $f$ is an injection.
From Surjection by Restriction of Codomain, $f: I \to J$ is a surjection.
{{qed}} | Let $f$ be a [[Definition:Real Function|real function]] which is defined on $I \subseteq \R$.
Let $f$ be [[Definition:Strictly Monotone Real Function|strictly monotone]] on $I$.
Let the [[Definition:Image of Mapping|image]] of $f$ be $J$.
Then $f: I \to J$ is a [[Definition:Bijection|bijection]]. | From [[Strictly Monotone Mapping with Totally Ordered Domain is Injective]], $f$ is an [[Definition:Injection|injection]].
From [[Surjection by Restriction of Codomain]], $f: I \to J$ is a [[Definition:Surjection|surjection]].
{{qed}} | Strictly Monotone Real Function is Bijective | https://proofwiki.org/wiki/Strictly_Monotone_Real_Function_is_Bijective | https://proofwiki.org/wiki/Strictly_Monotone_Real_Function_is_Bijective | [
"Strictly Monotone Real Functions",
"Bijections"
] | [
"Definition:Real Function",
"Definition:Strictly Monotone/Real Function",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Bijection"
] | [
"Strictly Monotone Mapping with Totally Ordered Domain is Injective",
"Definition:Injection",
"Restriction of Mapping to Image is Surjection",
"Definition:Surjection"
] |
proofwiki-1629 | Inverse of Strictly Monotone Function | Let $f$ be a real function which is defined on $I \subseteq \R$.
Let $f$ be strictly monotone on $I$.
Let the image of $f$ be $J$.
Then $f$ always has an inverse function $f^{-1}$ and:
:if $f$ is strictly increasing then so is $f^{-1}$
:if $f$ is strictly decreasing then so is $f^{-1}$. | The function $f$ is a bijection from Strictly Monotone Real Function is Bijective.
Hence from Bijection iff Inverse is Bijection, $f^{-1}$ always exists and is also a bijection.
From the definition of strictly increasing:
:$X < Y \iff \map f X < \map f Y$
Hence, substituting $X = \map {f^{-1}} x$ and $Y = \map {f^{-1}}... | Let $f$ be a [[Definition:Real Function|real function]] which is defined on $I \subseteq \R$.
Let $f$ be [[Definition:Strictly Monotone Real Function|strictly monotone]] on $I$.
Let the [[Definition:Image of Mapping|image]] of $f$ be $J$.
Then $f$ always has an [[Definition:Inverse Mapping|inverse function]] $f^{-... | The function $f$ is a [[Definition:Bijection|bijection]] from [[Strictly Monotone Real Function is Bijective]].
Hence from [[Bijection iff Inverse is Bijection]], $f^{-1}$ always exists and is also a [[Definition:Bijection|bijection]].
From the definition of [[Definition:Strictly Increasing Real Function|strictly in... | Inverse of Strictly Monotone Function | https://proofwiki.org/wiki/Inverse_of_Strictly_Monotone_Function | https://proofwiki.org/wiki/Inverse_of_Strictly_Monotone_Function | [
"Strictly Monotone Real Functions",
"Inverse Mappings"
] | [
"Definition:Real Function",
"Definition:Strictly Monotone/Real Function",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Inverse Mapping",
"Definition:Strictly Increasing/Real Function",
"Definition:Strictly Decreasing/Real Function"
] | [
"Definition:Bijection",
"Strictly Monotone Real Function is Bijective",
"Inverse of Bijection is Bijection",
"Definition:Bijection",
"Definition:Strictly Increasing/Real Function",
"Definition:Substitution",
"Definition:Strictly Decreasing/Real Function",
"Definition:Substitution"
] |
proofwiki-1630 | Rokhlin's Theorem (Intersection Forms) | Let $M$ be a smooth 4-manifold.
Then:
:$\map {\omega_2} {\map T M} = 0 \implies \operatorname {sign} Q_M = 0 \pmod {16}$
where:
:$Q_M$ is the intersection form
:$\map T M$ is the tangent bundle
:$\omega_2$ is the second Stiefel-Whitney class. | {{Explain|What does "sign" mean in this context?}}
{{ProofWanted}}
{{Namedfor|Vladimir Abramovich Rokhlin|cat = Rokhlin}}
Category:Smooth Manifolds
e3ky606b3n719em4ptt4lryvu2cn6uk | Let $M$ be a [[Definition:Smooth Manifold|smooth 4-manifold]].
Then:
:$\map {\omega_2} {\map T M} = 0 \implies \operatorname {sign} Q_M = 0 \pmod {16}$
where:
:$Q_M$ is the [[Definition:Intersection Form|intersection form]]
:$\map T M$ is the [[Definition:Tangent Bundle|tangent bundle]]
:$\omega_2$ is the second [[Def... | {{Explain|What does "sign" mean in this context?}}
{{ProofWanted}}
{{Namedfor|Vladimir Abramovich Rokhlin|cat = Rokhlin}}
[[Category:Smooth Manifolds]]
e3ky606b3n719em4ptt4lryvu2cn6uk | Rokhlin's Theorem (Intersection Forms) | https://proofwiki.org/wiki/Rokhlin's_Theorem_(Intersection_Forms) | https://proofwiki.org/wiki/Rokhlin's_Theorem_(Intersection_Forms) | [
"Smooth Manifolds"
] | [
"Definition:Topological Manifold/Smooth Manifold",
"Definition:Intersection Form",
"Definition:Tangent Bundle",
"Definition:Stiefel-Whitney Class"
] | [
"Category:Smooth Manifolds"
] |
proofwiki-1631 | Convex Real Function is Continuous | Let $f$ be a real function which is convex on the open interval $\openint a b$.
Then $f$ is continuous on $\openint a b$. | From Convex Real Function is Left-Hand and Right-Hand Differentiable, $f$ is left-hand and right-hand differentiable on $\openint a b$.
From Left-Hand and Right-Hand Differentiable Function is Continuous, $f$ is continuous on $\openint a b$.
{{qed}} | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Convex Real Function|convex]] on the [[Definition:Open Real Interval|open interval]] $\openint a b$.
Then $f$ is [[Definition:Continuous on Interval|continuous]] on $\openint a b$. | From [[Convex Real Function is Left-Hand and Right-Hand Differentiable]], $f$ is [[Definition:Left-Hand Derivative|left-hand]] and [[Definition:Right-Hand Derivative|right-hand differentiable]] on $\openint a b$.
From [[Left-Hand and Right-Hand Differentiable Function is Continuous]], $f$ is [[Definition:Continuous on... | Convex Real Function is Continuous | https://proofwiki.org/wiki/Convex_Real_Function_is_Continuous | https://proofwiki.org/wiki/Convex_Real_Function_is_Continuous | [
"Convex Real Functions",
"Continuous Real Functions"
] | [
"Definition:Real Function",
"Definition:Convex Real Function",
"Definition:Real Interval/Open",
"Definition:Continuous Real Function/Interval"
] | [
"Convex Real Function is Left-Hand and Right-Hand Differentiable",
"Definition:Left-Hand Derivative",
"Definition:Right-Hand Derivative",
"Left-Hand and Right-Hand Differentiable Function is Continuous",
"Definition:Continuous Real Function/Interval"
] |
proofwiki-1632 | Convex Real Function is Left-Hand and Right-Hand Differentiable | Let $f$ be a real function which is convex on the open interval $\openint a b$.
Then the left-hand derivative $\map {f'_-} x$ and right-hand derivative $\map {f'_+} x$ both exist for all $x \in \openint a b$. | Let $f$ be convex on $\openint a b$.
Then by definition of convexity:
:$\forall x_1, x_2, x_3 \in \openint a b: x_1 < x_2 < x_3: \dfrac {\map f {x_2} - \map f {x_1} } {x_2 - x_1} \le \dfrac {\map f {x_3} - \map f {x_1} } {x_3 - x_1}$
Let $0 < h_1 < h_2$.
Substitute $x_1 = x$, $x_2 = x + h_1$, $x_3 = x + h_2$. Then:
:$\... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Convex Real Function|convex]] on the [[Definition:Open Real Interval|open interval]] $\openint a b$.
Then the [[Definition:Left-Hand Derivative|left-hand derivative]] $\map {f'_-} x$ and [[Definition:Right-Hand Derivative|right-hand derivat... | Let $f$ be [[Definition:Convex Real Function|convex]] on $\openint a b$.
Then by definition of [[Definition:Convex Real Function|convexity]]:
:$\forall x_1, x_2, x_3 \in \openint a b: x_1 < x_2 < x_3: \dfrac {\map f {x_2} - \map f {x_1} } {x_2 - x_1} \le \dfrac {\map f {x_3} - \map f {x_1} } {x_3 - x_1}$
Let $0 < h... | Convex Real Function is Left-Hand and Right-Hand Differentiable | https://proofwiki.org/wiki/Convex_Real_Function_is_Left-Hand_and_Right-Hand_Differentiable | https://proofwiki.org/wiki/Convex_Real_Function_is_Left-Hand_and_Right-Hand_Differentiable | [
"Convex Real Functions"
] | [
"Definition:Real Function",
"Definition:Convex Real Function",
"Definition:Real Interval/Open",
"Definition:Left-Hand Derivative",
"Definition:Right-Hand Derivative"
] | [
"Definition:Convex Real Function",
"Definition:Convex Real Function",
"Definition:Increasing/Real Function",
"Limit of Monotone Real Function/Increasing"
] |
proofwiki-1633 | Real Function is Convex iff Derivative is Increasing | Let $f$ be a real function which is differentiable on the open interval $\openint a b$.
Then:
:$f$ is convex on $\openint a b$
{{iff}}:
:its derivative $f'$ is increasing on $\openint a b$.
Thus the intuitive result that a convex function "gets steeper". | === Necessary Condition ===
Let $f$ be convex on $\openint a b$.
Let $x_1, x_2, x_3, x_4 \in \openint a b$ such that:
:$x_1 < x_2 < x_3 < x_4$
By the definition of convex function:
:$\dfrac {\map f {x_2} - \map f {x_1} } {x_2 - x_1} \le \dfrac {\map f {x_3} - \map f {x_2} } {x_3 - x_2} \le \dfrac {\map f {x_4} - \map f... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Differentiable on Interval|differentiable]] on the [[Definition:Open Real Interval|open interval]] $\openint a b$.
Then:
:$f$ is [[Definition:Convex Real Function|convex]] on $\openint a b$
{{iff}}:
:its [[Definition:Derivative on Interval|... | === Necessary Condition ===
Let $f$ be [[Definition:Convex Real Function|convex]] on $\openint a b$.
Let $x_1, x_2, x_3, x_4 \in \openint a b$ such that:
:$x_1 < x_2 < x_3 < x_4$
By the definition of [[Definition:Convex Real Function|convex function]]:
:$\dfrac {\map f {x_2} - \map f {x_1} } {x_2 - x_1} \le \dfrac ... | Real Function is Convex iff Derivative is Increasing | https://proofwiki.org/wiki/Real_Function_is_Convex_iff_Derivative_is_Increasing | https://proofwiki.org/wiki/Real_Function_is_Convex_iff_Derivative_is_Increasing | [
"Differential Calculus",
"Convex Real Functions"
] | [
"Definition:Real Function",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Real Interval/Open",
"Definition:Convex Real Function",
"Definition:Derivative/Real Function/Derivative on Interval",
"Definition:Increasing/Real Function",
"Definition:Convex Real Function"
] | [
"Definition:Convex Real Function",
"Definition:Convex Real Function",
"Definition:Increasing/Real Function",
"Definition:Increasing/Real Function",
"Definition:Increasing/Real Function",
"Definition:Convex Real Function"
] |
proofwiki-1634 | Inverse of Strictly Increasing Convex Real Function is Concave | Let $f$ be a real function which is convex on the open interval $I$.
Let $J = f \sqbrk I$.
If $f$ be strictly increasing on $I$, then $f^{-1}$ is concave on $J$. | Let:
:$X = \map f x \in J$
:$Y = \map f y \in J$.
From the definition of convex:
:$\forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: \map f {\alpha x + \beta y} \le \alpha \map f x + \beta \map f y$
Let $f$ be strictly increasing on $I$.
From Inverse of Strictly Monotone Function it follows that, $f^{-1}$ is stric... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Convex Real Function|convex]] on the [[Definition:Open Real Interval|open interval]] $I$.
Let $J = f \sqbrk I$.
If $f$ be [[Definition:Strictly Increasing Real Function|strictly increasing]] on $I$, then $f^{-1}$ is [[Definition:Concave Re... | Let:
:$X = \map f x \in J$
:$Y = \map f y \in J$.
From the definition of [[Definition:Convex Real Function|convex]]:
:$\forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: \map f {\alpha x + \beta y} \le \alpha \map f x + \beta \map f y$
Let $f$ be [[Definition:Strictly Increasing Real Function|strictly increasing... | Inverse of Strictly Increasing Convex Real Function is Concave | https://proofwiki.org/wiki/Inverse_of_Strictly_Increasing_Convex_Real_Function_is_Concave | https://proofwiki.org/wiki/Inverse_of_Strictly_Increasing_Convex_Real_Function_is_Concave | [
"Convex Real Functions",
"Concave Real Functions",
"Strictly Increasing Real Functions"
] | [
"Definition:Real Function",
"Definition:Convex Real Function",
"Definition:Real Interval/Open",
"Definition:Strictly Increasing/Real Function",
"Definition:Concave Real Function"
] | [
"Definition:Convex Real Function",
"Definition:Strictly Increasing/Real Function",
"Inverse of Strictly Monotone Function",
"Definition:Strictly Increasing/Real Function",
"Definition:Concave Real Function"
] |
proofwiki-1635 | Darboux's Theorem | Let $f$ be a real function which is bounded on the closed interval $\closedint a b$.
Then:
:$\ds m \paren {b - a} \le \underline {\int_a^b} \map f x \rd x \le \overline {\int_a^b} \map f x \rd x \le M \paren {b - a}$
where:
:$M$ is an upper bound of $f$ on $\closedint a b$
:$m$ is a lower bound of $f$ on $\closedint a ... | Consider the finite subdivision of $\closedint a b$ given by:
:$P = \set {a, b}$
From their definitions, the upper and lower Darboux sums can be computed as:
:$\ds \map {U^{\paren f}} P = \paren {b - a} \sup_{x \in \closedint a b} \map f x$
:$\ds \map {L^{\paren f}} P = \paren {b - a} \inf_{x \in \closedint a b} \map f... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Bounded Real-Valued Function|bounded]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$.
Then:
:$\ds m \paren {b - a} \le \underline {\int_a^b} \map f x \rd x \le \overline {\int_a^b} \map f x \rd x \le M \paren {b... | Consider the [[Definition:Finite Subdivision|finite subdivision]] of $\closedint a b$ given by:
:$P = \set {a, b}$
From their definitions, the [[Definition:Upper Darboux Sum|upper]] and [[Definition:Lower Darboux Sum|lower Darboux sums]] can be computed as:
:$\ds \map {U^{\paren f}} P = \paren {b - a} \sup_{x \in \clo... | Darboux's Theorem | https://proofwiki.org/wiki/Darboux's_Theorem | https://proofwiki.org/wiki/Darboux's_Theorem | [
"Darboux's Theorem",
"Integral Calculus"
] | [
"Definition:Real Function",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Real Interval/Closed",
"Definition:Upper Bound of Mapping/Real-Valued",
"Definition:Lower Bound of Mapping/Real-Valued",
"Definition:Upper Darboux Integral",
"Definition:Lower Darboux Integral",
"Definition:Darboux Integ... | [
"Definition:Subdivision of Interval/Finite",
"Definition:Upper Darboux Sum",
"Definition:Lower Darboux Sum",
"Definition:Supremum of Mapping/Real-Valued Function",
"Definition:Infimum of Mapping/Real-Valued Function",
"Definition:Upper Darboux Integral",
"Definition:Lower Darboux Integral",
"Upper Dar... |
proofwiki-1636 | Wilson's Theorem | A (strictly) positive integer $p$ is a prime {{iff}}:
:$\paren {p - 1}! \equiv -1 \pmod p$ | If $p = 2$ the result is immediate.
Therefore we assume that $p$ is an odd prime. | A [[Definition:Strictly Positive Integer|(strictly) positive integer]] $p$ is a [[Definition:Prime Number|prime]] {{iff}}:
:$\paren {p - 1}! \equiv -1 \pmod p$ | If $p = 2$ the result is immediate.
Therefore we assume that $p$ is an [[Definition:Odd Prime|odd prime]]. | Wilson's Theorem | https://proofwiki.org/wiki/Wilson's_Theorem | https://proofwiki.org/wiki/Wilson's_Theorem | [
"Wilson's Theorem",
"Prime Numbers",
"Factorials",
"Modulo Arithmetic"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Prime Number"
] | [
"Definition:Odd Prime"
] |
proofwiki-1637 | Wilson's Theorem | A (strictly) positive integer $p$ is a prime {{iff}}:
:$\paren {p - 1}! \equiv -1 \pmod p$ | Proof by induction:
Let $\map P n$ be the proposition:
:$\dfrac {n!} {p^\mu} \equiv \paren {-1}^\mu a_0! a_1! \dotsm a_k! \pmod p$
where $p, a_0, \dots, a_k, \mu$ are as defined above.
=== Basis for the Induction ===
For $n = 1$:
:$a_0 = 1, \mu = 0$
$\map P 1$ reduces to:
:$\dfrac {1!} {p^0} \equiv \paren {-1}^0 1! \pm... | A [[Definition:Strictly Positive Integer|(strictly) positive integer]] $p$ is a [[Definition:Prime Number|prime]] {{iff}}:
:$\paren {p - 1}! \equiv -1 \pmod p$ | Proof by [[Principle of Mathematical Induction|induction]]:
Let $\map P n$ be the proposition:
:$\dfrac {n!} {p^\mu} \equiv \paren {-1}^\mu a_0! a_1! \dotsm a_k! \pmod p$
where $p, a_0, \dots, a_k, \mu$ are as defined above.
=== Basis for the Induction ===
For $n = 1$:
:$a_0 = 1, \mu = 0$
$\map P 1$ reduces to:
:... | Wilson's Theorem/Corollary 2/Proof 1 | https://proofwiki.org/wiki/Wilson's_Theorem | https://proofwiki.org/wiki/Wilson's_Theorem/Corollary_2/Proof_1 | [
"Wilson's Theorem",
"Prime Numbers",
"Factorials",
"Modulo Arithmetic"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Prime Number"
] | [
"Principle of Mathematical Induction",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Definition:Number Base",
"Definition:Induction Step",
"Definition:Power (Algebra)",
"Wilson's Theorem/Corollary 2/Proof 1",
"Wilson's Theorem",
"Factorial/Examples/0",
"Principle of Mat... |
proofwiki-1638 | Wilson's Theorem | A (strictly) positive integer $p$ is a prime {{iff}}:
:$\paren {p - 1}! \equiv -1 \pmod p$ | Consider the numbers of $\set {1, 2, \ldots, n}$ which are not multiples of $p$.
There are $\floor {\dfrac n p}$ complete sets of $p - 1$ such consecutive elements of $\set {1, 2, \ldots, n}$.
Each one of these has a product which is congruent to $-1 \pmod p$ by Wilson's Theorem.
There are also $a_0$ left over which ar... | A [[Definition:Strictly Positive Integer|(strictly) positive integer]] $p$ is a [[Definition:Prime Number|prime]] {{iff}}:
:$\paren {p - 1}! \equiv -1 \pmod p$ | Consider the numbers of $\set {1, 2, \ldots, n}$ which are not [[Definition:Integer Multiple|multiples]] of $p$.
There are $\floor {\dfrac n p}$ complete sets of $p - 1$ such consecutive elements of $\set {1, 2, \ldots, n}$.
Each one of these has a product which is [[Definition:Congruence Modulo Integer|congruent]] t... | Wilson's Theorem/Corollary 2/Proof 2 | https://proofwiki.org/wiki/Wilson's_Theorem | https://proofwiki.org/wiki/Wilson's_Theorem/Corollary_2/Proof_2 | [
"Wilson's Theorem",
"Prime Numbers",
"Factorials",
"Modulo Arithmetic"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Prime Number"
] | [
"Definition:Integral Multiple/Real Numbers",
"Definition:Congruence (Number Theory)/Integers",
"Wilson's Theorem",
"Definition:Congruence (Number Theory)/Integers",
"Definition:Divisor (Algebra)/Integer",
"Definition:Integral Multiple/Real Numbers",
"Definition:Divisor (Algebra)/Integer",
"Definition:... |
proofwiki-1639 | Wilson's Theorem | A (strictly) positive integer $p$ is a prime {{iff}}:
:$\paren {p - 1}! \equiv -1 \pmod p$ | If $p = 2$ the result is obvious.
Therefore we assume that $p$ is an odd prime.
Let $p$ be prime.
Consider $n \in \Z, 1 \le n < p$.
As $p$ is prime, $n \perp p$.
From Law of Inverses (Modulo Arithmetic), we have:
:$\exists n' \in \Z, 1 \le n' < p: n n' \equiv 1 \pmod p$
By Solution of Linear Congruence, for each $n$ th... | A [[Definition:Strictly Positive Integer|(strictly) positive integer]] $p$ is a [[Definition:Prime Number|prime]] {{iff}}:
:$\paren {p - 1}! \equiv -1 \pmod p$ | If $p = 2$ the result is obvious.
Therefore we assume that $p$ is an [[Definition:Odd Prime|odd prime]].
Let $p$ be [[Definition:Prime Number|prime]].
Consider $n \in \Z, 1 \le n < p$.
As $p$ is [[Definition:Prime Number|prime]], $n \perp p$.
From [[Law of Inverses (Modulo Arithmetic)]], we have:
:$\exists n' \i... | Wilson's Theorem/Necessary Condition/Proof 1 | https://proofwiki.org/wiki/Wilson's_Theorem | https://proofwiki.org/wiki/Wilson's_Theorem/Necessary_Condition/Proof_1 | [
"Wilson's Theorem",
"Prime Numbers",
"Factorials",
"Modulo Arithmetic"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Prime Number"
] | [
"Definition:Odd Prime",
"Definition:Prime Number",
"Definition:Prime Number",
"Law of Inverses (Modulo Arithmetic)",
"Solution of Linear Congruence",
"Difference of Two Squares",
"Definition:Odd Prime",
"Negative Number is Congruent to Modulus minus Number",
"Definition:Congruence (Number Theory)/In... |
proofwiki-1640 | Wilson's Theorem | A (strictly) positive integer $p$ is a prime {{iff}}:
:$\paren {p - 1}! \equiv -1 \pmod p$ | If $p = 2$ the result is obvious.
Therefore we assume that $p$ is an odd prime.
Consider $p$ points on the circumference of a circle $C$ dividing it into $p$ equal arcs.
By joining these points in a cycle, we can create polygons, some of them stellated.
From Number of Different n-gons that can be Inscribed in Circle, t... | A [[Definition:Strictly Positive Integer|(strictly) positive integer]] $p$ is a [[Definition:Prime Number|prime]] {{iff}}:
:$\paren {p - 1}! \equiv -1 \pmod p$ | If $p = 2$ the result is obvious.
Therefore we assume that $p$ is an [[Definition:Odd Prime|odd prime]].
Consider $p$ [[Definition:Point|points]] on the [[Definition:Circumference of Circle|circumference]] of a [[Definition:Circle|circle]] $C$ dividing it into $p$ equal [[Definition:Arc of Circle|arcs]].
By joining... | Wilson's Theorem/Necessary Condition/Proof 2 | https://proofwiki.org/wiki/Wilson's_Theorem | https://proofwiki.org/wiki/Wilson's_Theorem/Necessary_Condition/Proof_2 | [
"Wilson's Theorem",
"Prime Numbers",
"Factorials",
"Modulo Arithmetic"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Prime Number"
] | [
"Definition:Odd Prime",
"Definition:Point",
"Definition:Circle/Circumference",
"Definition:Circle",
"Definition:Circle/Arc",
"Definition:Point",
"Definition:Cyclic Permutation",
"Definition:Polygon",
"Definition:Stellation/Polygon",
"Number of Different n-gons that can be Inscribed in Circle",
"... |
proofwiki-1641 | Wilson's Theorem | A (strictly) positive integer $p$ is a prime {{iff}}:
:$\paren {p - 1}! \equiv -1 \pmod p$ | Let $p$ be prime.
Consider $\struct {\Z_p, +, \times}$, the ring of integers modulo $m$.
From Ring of Integers Modulo Prime is Field, $\struct {\Z_p, +, \times}$ is a field.
Hence, apart from $\eqclass 0 p$, all elements of $\struct {\Z_p, +, \times}$ are units
As $\struct {\Z_p, +, \times}$ is a field, it is also by d... | A [[Definition:Strictly Positive Integer|(strictly) positive integer]] $p$ is a [[Definition:Prime Number|prime]] {{iff}}:
:$\paren {p - 1}! \equiv -1 \pmod p$ | Let $p$ be [[Definition:Prime Number|prime]].
Consider $\struct {\Z_p, +, \times}$, the [[Definition:Ring of Integers Modulo m|ring of integers modulo $m$]].
From [[Ring of Integers Modulo Prime is Field]], $\struct {\Z_p, +, \times}$ is a [[Definition:Field (Abstract Algebra)|field]].
Hence, apart from $\eqclass 0 ... | Wilson's Theorem/Necessary Condition/Proof 3 | https://proofwiki.org/wiki/Wilson's_Theorem | https://proofwiki.org/wiki/Wilson's_Theorem/Necessary_Condition/Proof_3 | [
"Wilson's Theorem",
"Prime Numbers",
"Factorials",
"Modulo Arithmetic"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Prime Number"
] | [
"Definition:Prime Number",
"Definition:Ring of Integers Modulo m",
"Ring of Integers Modulo Prime is Field",
"Definition:Field (Abstract Algebra)",
"Definition:Element",
"Definition:Unit of Ring",
"Definition:Field (Abstract Algebra)",
"Definition:Integral Domain",
"Product of Units of Integral Doma... |
proofwiki-1642 | Integral of Constant | Let $c$ be a constant. | Let $F$ be a primitive of $f$ on $\closedint a b$.
By Primitive of Constant Multiple of Function, $H = c F$ is a primitive of $c f$ on $\closedint a b$.
Hence by the Fundamental Theorem of Calculus:
{{begin-eqn}}
{{eqn | l = \int_a^b c \map f x \rd x
| r = \bigintlimits {c \map F x} a b
| c =
}}
{{eqn | r ... | Let $c$ be a [[Definition:Constant|constant]]. | Let $F$ be a [[Definition:Primitive (Calculus)|primitive]] of $f$ on $\closedint a b$.
By [[Primitive of Constant Multiple of Function]], $H = c F$ is a [[Definition:Primitive (Calculus)|primitive]] of $c f$ on $\closedint a b$.
Hence by the [[Fundamental Theorem of Calculus]]:
{{begin-eqn}}
{{eqn | l = \int_a^b c \... | Definite Integral of Constant Multiple of Real Function/Proof 1 | https://proofwiki.org/wiki/Integral_of_Constant | https://proofwiki.org/wiki/Definite_Integral_of_Constant_Multiple_of_Real_Function/Proof_1 | [
"Integral Calculus"
] | [
"Definition:Constant"
] | [
"Definition:Primitive (Calculus)",
"Primitive of Constant Multiple of Function",
"Definition:Primitive (Calculus)",
"Fundamental Theorem of Calculus"
] |
proofwiki-1643 | Integral of Constant | Let $c$ be a constant. | Let $F$ be a primitive of $f$ on $\closedint a b$.
By Linear Combination of Definite Integrals:
:$\ds \int_a^b \paren {\lambda \map f t + \mu \map g t} \rd t = \lambda \int_a^b \map f t \rd t + \mu \int_a^b \map g t \rd t$
for real functions $f$ and $g$ which are integrable on the closed interval $\closedint a b$, wher... | Let $c$ be a [[Definition:Constant|constant]]. | Let $F$ be a [[Definition:Primitive (Calculus)|primitive]] of $f$ on $\closedint a b$.
By [[Linear Combination of Definite Integrals]]:
:$\ds \int_a^b \paren {\lambda \map f t + \mu \map g t} \rd t = \lambda \int_a^b \map f t \rd t + \mu \int_a^b \map g t \rd t$
for [[Definition:Real Function|real functions]] $f$ an... | Definite Integral of Constant Multiple of Real Function/Proof 2 | https://proofwiki.org/wiki/Integral_of_Constant | https://proofwiki.org/wiki/Definite_Integral_of_Constant_Multiple_of_Real_Function/Proof_2 | [
"Integral Calculus"
] | [
"Definition:Constant"
] | [
"Definition:Primitive (Calculus)",
"Linear Combination of Integrals/Definite",
"Definition:Real Function",
"Definition:Integrable Function",
"Definition:Real Interval/Closed",
"Definition:Real Number"
] |
proofwiki-1644 | Sum of Integrals on Adjacent Intervals for Continuous Functions | Let $f$ be a real function which is continuous on any closed interval $I$.
Let $a, b, c \in I$.
Then:
:$\ds \int_a^c \map f t \rd t + \int_c^b \map f t \rd t = \int_a^b \map f t \rd t$ | By Continuous Real Function is Darboux Integrable, $f$ is integrable on $I$.
The result follows by application of Sum of Integrals on Adjacent Intervals for Integrable Functions.
{{qed}} | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Continuous on Interval|continuous]] on any [[Definition:Closed Real Interval|closed interval]] $I$.
Let $a, b, c \in I$.
Then:
:$\ds \int_a^c \map f t \rd t + \int_c^b \map f t \rd t = \int_a^b \map f t \rd t$ | By [[Continuous Real Function is Darboux Integrable]], $f$ is [[Definition:Definite Integral|integrable]] on $I$.
The result follows by application of [[Sum of Integrals on Adjacent Intervals for Integrable Functions]].
{{qed}} | Sum of Integrals on Adjacent Intervals for Continuous Functions | https://proofwiki.org/wiki/Sum_of_Integrals_on_Adjacent_Intervals_for_Continuous_Functions | https://proofwiki.org/wiki/Sum_of_Integrals_on_Adjacent_Intervals_for_Continuous_Functions | [
"Integral Calculus"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Interval",
"Definition:Real Interval/Closed"
] | [
"Continuous Real Function is Darboux Integrable",
"Definition:Definite Integral",
"Sum of Integrals on Adjacent Intervals for Integrable Functions"
] |
proofwiki-1645 | Primitives which Differ by Constant | Let $F$ be a primitive for a real function $f$ on the closed interval $\closedint a b$.
Let $G$ be a real function defined on $\closedint a b$.
Then $G$ is a primitive for $f$ on $\closedint a b$ {{iff}}:
:$\exists c \in \R: \forall x \in \closedint a b: \map G x = \map F x + c$
That is, {{iff}} $F$ and $G$ differ by a... | === Necessary Condition ===
Suppose $G$ is a primitive for $f$.
Then $F - G$ is continuous on $\closedint a b$, differentiable on $\openint a b$, and for any $x \in \openint a b$, we have:
{{begin-eqn}}
{{eqn | l = \map {D_x} {\map F x - \map G x}
| r = \map {D_x} {\map F x} - \map {D_x} {\map G x}
| c = Su... | Let $F$ be a [[Definition:Primitive (Calculus)|primitive]] for a [[Definition:Real Function|real function]] $f$ on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$.
Let $G$ be a [[Definition:Real Function|real function]] defined on $\closedint a b$.
Then $G$ is a [[Definition:Primitive (Calcu... | === Necessary Condition ===
Suppose $G$ is a [[Definition:Primitive (Calculus)|primitive]] for $f$.
Then $F - G$ is [[Definition:Continuous on Interval|continuous]] on $\closedint a b$, [[Definition:Differentiable on Interval|differentiable]] on $\openint a b$, and for any $x \in \openint a b$, we have:
{{begin-eqn}... | Primitives which Differ by Constant | https://proofwiki.org/wiki/Primitives_which_Differ_by_Constant | https://proofwiki.org/wiki/Primitives_which_Differ_by_Constant | [
"Primitives which Differ by Constant",
"Primitives"
] | [
"Definition:Primitive (Calculus)",
"Definition:Real Function",
"Definition:Real Interval/Closed",
"Definition:Real Function",
"Definition:Primitive (Calculus)",
"Definition:Primitive (Calculus)/Constant of Integration",
"Definition:Real Interval/Closed"
] | [
"Definition:Primitive (Calculus)",
"Definition:Continuous Real Function/Interval",
"Definition:Differentiable Mapping/Real Function/Interval",
"Sum Rule for Derivatives",
"Definition:Primitive (Calculus)",
"Zero Derivative implies Constant Function",
"Definition:Primitive (Calculus)/Constant of Integrat... |
proofwiki-1646 | Definite Integral of Function plus Constant | Let $f$ be a real function which is continuous on the closed interval $\closedint a b$.
Let $c$ be a constant.
Then:
:$\ds \int_a^b \paren {\map f t + c} \rd t = \int_a^b \map f t \rd t + c \paren {b - a}$ | Let $P = \set {x_0, x_1, x_2, \ldots, x_n}$ be a finite subdivision of $\closedint a b$.
Let $\map {L^{\paren {f + c} } } P$ be the lower Darboux sum of $\map f x + c$ on $\closedint a b$ belonging to $P$.
Let:
:$\ds m_k^{\paren {f + c} } = \map {\inf_{x \mathop \in \closedint {x_{k - 1} } {x_k} } } {\map f x + c}$
whe... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Continuous on Interval|continuous]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$.
Let $c$ be a constant.
Then:
:$\ds \int_a^b \paren {\map f t + c} \rd t = \int_a^b \map f t \rd t + c \paren {b - a}$ | Let $P = \set {x_0, x_1, x_2, \ldots, x_n}$ be a [[Definition:Finite Subdivision|finite subdivision]] of $\closedint a b$.
Let $\map {L^{\paren {f + c} } } P$ be the [[Definition:Lower Darboux Sum|lower Darboux sum]] of $\map f x + c$ on $\closedint a b$ belonging to $P$.
Let:
:$\ds m_k^{\paren {f + c} } = \map {\inf... | Definite Integral of Function plus Constant | https://proofwiki.org/wiki/Definite_Integral_of_Function_plus_Constant | https://proofwiki.org/wiki/Definite_Integral_of_Function_plus_Constant | [
"Integral Calculus"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Interval",
"Definition:Real Interval/Closed"
] | [
"Definition:Subdivision of Interval/Finite",
"Definition:Lower Darboux Sum",
"Telescoping Series/Example 2",
"Definition:Definite Integral"
] |
proofwiki-1647 | Continuous Real Function is Darboux Integrable | Let $f$ be a real function which is continuous on the closed interval $\closedint a b$.
Then $f$ is Darboux integrable on $\closedint a b$. | We have the result $f$ is bounded by Continuous Real Function is Bounded.
By Condition for Darboux Integrability, it suffices to show that for all $\epsilon > 0$, there exists a subdivision $P$ of $\closedint a b$ such that:
:$\map U P - \map L P < \epsilon$
where $\map U P$ and $\map L P$ denote the upper Darboux sum ... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Continuous on Interval|continuous]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$.
Then $f$ is [[Definition:Darboux Integrable Function|Darboux integrable]] on $\closedint a b$. | We have the result $f$ is [[Definition:Bounded Real-Valued Function|bounded]] by [[Continuous Real Function is Bounded]].
By [[Condition for Darboux Integrability]], it suffices to show that for all $\epsilon > 0$, there exists a [[Definition:Subdivision of Interval|subdivision]] $P$ of $\closedint a b$ such that:
:$\... | Continuous Real Function is Darboux Integrable | https://proofwiki.org/wiki/Continuous_Real_Function_is_Darboux_Integrable | https://proofwiki.org/wiki/Continuous_Real_Function_is_Darboux_Integrable | [
"Darboux Integrable Functions",
"Integral Calculus",
"Continuous Real Functions"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Interval",
"Definition:Real Interval/Closed",
"Definition:Darboux Integrable Function"
] | [
"Definition:Bounded Mapping/Real-Valued",
"Image of Closed Real Interval is Bounded",
"Condition for Darboux Integrability",
"Definition:Subdivision of Interval",
"Definition:Upper Darboux Sum",
"Definition:Lower Darboux Sum",
"Definition:Subdivision of Interval",
"Continuous Function on Closed Real I... |
proofwiki-1648 | Definite Integral on Zero Interval | Let $f$ be a real function which is defined on the closed interval $\Bbb I := \closedint a b$, where $a < b$.
Then:
:$\ds \forall c \in \Bbb I: \int_c^c \map f t \rd t = 0$ | Follows directly from the definition of definite integral.
There is only one finite subdivision of $\closedint c c$ and that is $\set c$.
Both the lower Darboux sum and upper Darboux sum of $\map f x$ on $\closedint c c$ belonging to the finite subdivision $\set c$ are equal to zero.
Hence the result.
{{qed}} | Let $f$ be a [[Definition:Real Function|real function]] which is defined on the [[Definition:Closed Real Interval|closed interval]] $\Bbb I := \closedint a b$, where $a < b$.
Then:
:$\ds \forall c \in \Bbb I: \int_c^c \map f t \rd t = 0$ | Follows directly from the definition of [[Definition:Definite Integral|definite integral]].
There is only one [[Definition:Finite Subdivision|finite subdivision]] of $\closedint c c$ and that is $\set c$.
Both the [[Definition:Lower Darboux Sum|lower Darboux sum]] and [[Definition:Upper Darboux Sum|upper Darboux sum]... | Definite Integral on Zero Interval | https://proofwiki.org/wiki/Definite_Integral_on_Zero_Interval | https://proofwiki.org/wiki/Definite_Integral_on_Zero_Interval | [
"Definite Integrals"
] | [
"Definition:Real Function",
"Definition:Real Interval/Closed"
] | [
"Definition:Definite Integral",
"Definition:Subdivision of Interval/Finite",
"Definition:Lower Darboux Sum",
"Definition:Upper Darboux Sum",
"Definition:Subdivision of Interval/Finite",
"Definition:Zero (Number)"
] |
proofwiki-1649 | Linear Combination of Derivatives | Let $\map f x, \map g x$ be real functions defined on the open interval $I$.
Let $\xi \in I$ be a point in $I$ at which both $f$ and $g$ are differentiable.
Then:
:$\map D {\lambda f + \mu g} = \lambda D f + \mu D g$
at the point $\xi$ for some $\lambda \in \R$ and $\mu \in \R$.
It follows from the definition of deriva... | {{tidy}}
Let $\xi \in I$ be a point in $I$ at which $f$ is differentiable. Then by definition, $f$ is differentiable at the point $\xi$ implies that the limit:
:$\ds \lim_{h \mathop \to 0} \frac {\map f {\xi + h} - \map f \xi} h$
exists.
Let $\xi \in I$ be a point in $I$ at which $g$ is differentiable. Then by defi... | Let $\map f x, \map g x$ be [[Definition:Real Function|real functions]] defined on the [[Definition:Open Real Interval|open interval]] $I$.
Let $\xi \in I$ be a point in $I$ at which both $f$ and $g$ are [[Definition:Differentiable Real Function at Point|differentiable]].
Then:
:$\map D {\lambda f + \mu g} = \lambda... | {{tidy}}
Let $\xi \in I$ be a point in $I$ at which $f$ is [[Definition:Differentiable Real Function at Point|differentiable]]. Then by definition, $f$ is [[Definition:Differentiable Real Function at Point|differentiable at the point $\xi$]] implies that the [[Definition:Limit of Real Function|limit]]:
:$\ds \lim_{h... | Linear Combination of Derivatives | https://proofwiki.org/wiki/Linear_Combination_of_Derivatives | https://proofwiki.org/wiki/Linear_Combination_of_Derivatives | [
"Differential Calculus"
] | [
"Definition:Real Function",
"Definition:Real Interval/Open",
"Definition:Differentiable Mapping/Real Function/Point",
"Definition:Derivative/Real Function/Derivative on Interval",
"Definition:Differentiable Mapping/Real Function/Interval"
] | [
"Definition:Differentiable Mapping/Real Function/Point",
"Definition:Differentiable Mapping/Real Function/Point",
"Definition:Limit of Real Function",
"Definition:Differentiable Mapping/Real Function/Point",
"Definition:Differentiable Mapping/Real Function/Point",
"Definition:Limit of Real Function",
"D... |
proofwiki-1650 | Linear Combination of Integrals | Let $f$ and $g$ be real functions which are integrable on the closed interval $\closedint a b$.
Let $\lambda$ and $\mu$ be real numbers.
Then the following results hold: | Let $F$ and $G$ be primitives of $f$ and $g$ respectively on $\closedint a b$.
By Linear Combination of Derivatives, $H = \lambda F + \mu G$ is a primitive of $\lambda f + \mu g$ on $\closedint a b$.
Hence by the Fundamental Theorem of Calculus:
{{begin-eqn}}
{{eqn | l = \int_a^b \paren {\lambda \map f t + \mu \map g t... | Let $f$ and $g$ be [[Definition:Real Function|real functions]] which are [[Definition:Integrable Function|integrable]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$.
Let $\lambda$ and $\mu$ be [[Definition:Real Number|real numbers]].
Then the following results hold: | Let $F$ and $G$ be [[Definition:Primitive (Calculus)|primitives]] of $f$ and $g$ respectively on $\closedint a b$.
By [[Linear Combination of Derivatives]], $H = \lambda F + \mu G$ is a [[Definition:Primitive (Calculus)|primitive]] of $\lambda f + \mu g$ on $\closedint a b$.
Hence by the [[Fundamental Theorem of Calc... | Linear Combination of Integrals/Definite/Proof 1 | https://proofwiki.org/wiki/Linear_Combination_of_Integrals | https://proofwiki.org/wiki/Linear_Combination_of_Integrals/Definite/Proof_1 | [
"Integral Calculus"
] | [
"Definition:Real Function",
"Definition:Integrable Function",
"Definition:Real Interval/Closed",
"Definition:Real Number"
] | [
"Definition:Primitive (Calculus)",
"Linear Combination of Derivatives",
"Definition:Primitive (Calculus)",
"Fundamental Theorem of Calculus"
] |
proofwiki-1651 | Linear Combination of Integrals | Let $f$ and $g$ be real functions which are integrable on the closed interval $\closedint a b$.
Let $\lambda$ and $\mu$ be real numbers.
Then the following results hold: | It is clear that for step functions $s$ and $t$:
{{handwaving|"It is clear that ..."}}
{{MissingLinks}}
:$\ds \int_a^b \lambda \map s x + \mu \map t x \rd x = \lambda \int_a^b \map s x \rd x + \mu \int_a^b \map t x \rd x$
Under any partition, the lower Darboux sums and upper Darboux sums of $f$ and $g$ are step functio... | Let $f$ and $g$ be [[Definition:Real Function|real functions]] which are [[Definition:Integrable Function|integrable]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$.
Let $\lambda$ and $\mu$ be [[Definition:Real Number|real numbers]].
Then the following results hold: | It is clear that for [[Definition:Step Function|step functions]] $s$ and $t$:
{{handwaving|"It is clear that ..."}}
{{MissingLinks}}
:$\ds \int_a^b \lambda \map s x + \mu \map t x \rd x = \lambda \int_a^b \map s x \rd x + \mu \int_a^b \map t x \rd x$
Under any partition, the [[Definition:Lower Darboux Sum|lower Darb... | Linear Combination of Integrals/Definite/Proof 2 | https://proofwiki.org/wiki/Linear_Combination_of_Integrals | https://proofwiki.org/wiki/Linear_Combination_of_Integrals/Definite/Proof_2 | [
"Integral Calculus"
] | [
"Definition:Real Function",
"Definition:Integrable Function",
"Definition:Real Interval/Closed",
"Definition:Real Number"
] | [
"Definition:Step Function",
"Definition:Lower Darboux Sum",
"Definition:Upper Darboux Sum",
"Definition:Step Function",
"Definition:Lower Darboux Sum",
"Definition:Upper Darboux Sum",
"Definition:Lower Darboux Sum",
"Definition:Upper Darboux Sum",
"Definition:Lower Darboux Sum",
"Definition:Upper ... |
proofwiki-1652 | Integration by Parts | Let $f$ and $g$ be real functions which are continuous on the closed interval $\closedint a b$.
Let $f$ and $g$ have primitives $F$ and $G$ respectively on $\closedint a b$.
Then:
=== Primitive ===
{{:Integration by Parts/Primitive}}
=== Definite Integral ===
{{:Integration by Parts/Definite Integral}}
This technique i... | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d t} } {\map F t \map G t}
| r = \map f t \map G t + \map F t \map g t
| c = Product Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \paren {\map f t \map G t + \map F t \map g t} \rd t
| r = \map F t \map G t
| c = Fundamental Theorem of... | Let $f$ and $g$ be [[Definition:Real Function|real functions]] which are [[Definition:Continuous Real Function on Closed Interval|continuous]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$.
Let $f$ and $g$ have [[Definition:Primitive (Calculus)|primitives]] $F$ and $G$ respectively on $\c... | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d t} } {\map F t \map G t}
| r = \map f t \map G t + \map F t \map g t
| c = [[Product Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \paren {\map f t \map G t + \map F t \map g t} \rd t
| r = \map F t \map G t
| c = [[Fundamental Theo... | Integration by Parts/Primitive/Proof 1 | https://proofwiki.org/wiki/Integration_by_Parts | https://proofwiki.org/wiki/Integration_by_Parts/Primitive/Proof_1 | [
"Integration by Parts",
"Integral Calculus",
"Named Theorems",
"Proof Techniques"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Closed Interval",
"Definition:Real Interval/Closed",
"Definition:Primitive (Calculus)",
"Integration by Parts/Primitive",
"Integration by Parts/Definite Integral"
] | [
"Product Rule for Derivatives",
"Fundamental Theorem of Calculus",
"Definition:Primitive (Calculus)/Integration",
"Linear Combination of Integrals/Indefinite"
] |
proofwiki-1653 | Integration by Parts | Let $f$ and $g$ be real functions which are continuous on the closed interval $\closedint a b$.
Let $f$ and $g$ have primitives $F$ and $G$ respectively on $\closedint a b$.
Then:
=== Primitive ===
{{:Integration by Parts/Primitive}}
=== Definite Integral ===
{{:Integration by Parts/Definite Integral}}
This technique i... | Let $\map u x$ and $\map v x$ be integrable functions defined on $\closedint a b$.
Then:
{{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {u v}
| r = u \dfrac {\d v} {\d x} + v \dfrac {\d u} {\d x}
| c = Product Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = v \dfrac {\d u} {\d x}
| r = \m... | Let $f$ and $g$ be [[Definition:Real Function|real functions]] which are [[Definition:Continuous Real Function on Closed Interval|continuous]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$.
Let $f$ and $g$ have [[Definition:Primitive (Calculus)|primitives]] $F$ and $G$ respectively on $\c... | Let $\map u x$ and $\map v x$ be [[Definition:Integrable Function|integrable functions]] defined on $\closedint a b$.
Then:
{{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {u v}
| r = u \dfrac {\d v} {\d x} + v \dfrac {\d u} {\d x}
| c = [[Product Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| ... | Integration by Parts/Primitive/Proof 2 | https://proofwiki.org/wiki/Integration_by_Parts | https://proofwiki.org/wiki/Integration_by_Parts/Primitive/Proof_2 | [
"Integration by Parts",
"Integral Calculus",
"Named Theorems",
"Proof Techniques"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Closed Interval",
"Definition:Real Interval/Closed",
"Definition:Primitive (Calculus)",
"Integration by Parts/Primitive",
"Integration by Parts/Definite Integral"
] | [
"Definition:Integrable Function",
"Product Rule for Derivatives",
"Definition:Primitive (Calculus)/Integration",
"Linear Combination of Integrals/Indefinite",
"Fundamental Theorem of Calculus"
] |
proofwiki-1654 | Relative Sizes of Definite Integrals | Let $f$ and $g$ be real functions which are continuous on the closed interval $\closedint a b$, where $a < b$.
If:
:$\forall t \in \closedint a b: \map f t \le \map g t$
then:
:$\ds \int_a^b \map f t \rd t \le \int_a^b \map g t \rd t$
Similarly, if:
:$\forall t \in \closedint a b: \map f t < \map g t$
then:
:$\ds \int_... | Suppose that $\forall t \in \closedint a b: \map f t \le \map g t$.
From the Fundamental Theorem of Calculus, $g - f$ has a primitive on $\closedint a b$.
Let $H$ be such a primitive.
Then:
:$\forall t \in \closedint a b: D_t \map H t = \map g t - \map f t \ge 0$
By Derivative of Monotone Function, it follows that $H$ ... | Let $f$ and $g$ be [[Definition:Real Function|real functions]] which are [[Definition:Continuous Real Function on Closed Interval|continuous]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$, where $a < b$.
If:
:$\forall t \in \closedint a b: \map f t \le \map g t$
then:
:$\ds \int_a^b \ma... | Suppose that $\forall t \in \closedint a b: \map f t \le \map g t$.
From the [[Fundamental Theorem of Calculus]], $g - f$ has a [[Definition:Primitive (Calculus)|primitive]] on $\closedint a b$.
Let $H$ be such a [[Definition:Primitive (Calculus)|primitive]].
Then:
:$\forall t \in \closedint a b: D_t \map H t = \map... | Relative Sizes of Definite Integrals | https://proofwiki.org/wiki/Relative_Sizes_of_Definite_Integrals | https://proofwiki.org/wiki/Relative_Sizes_of_Definite_Integrals | [
"Integral Calculus",
"Inequalities"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Closed Interval",
"Definition:Real Interval/Closed"
] | [
"Fundamental Theorem of Calculus",
"Definition:Primitive (Calculus)",
"Definition:Primitive (Calculus)",
"Derivative of Monotone Function",
"Definition:Increasing/Real Function"
] |
proofwiki-1655 | Triangle Inequality for Integrals/Real | Let $f$ be a real function which is continuous on the closed interval $\closedint a b$.
Then:
:$\ds \size {\int_a^b \map f t \rd t} \le \int_a^b \size {\map f t} \rd t$ | From Negative of Absolute Value, we have for all $a \in \closedint a b$:
:$-\size {\map f t} \le \map f t \le \size {\map f t}$
Thus from Relative Sizes of Definite Integrals:
:$\ds -\int_a^b \size {\map f t} \rd t \le \int_a^b \map f t \rd t \le \int_a^b \size {\map f t} \rd t$
Hence the result.
{{qed}} | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Continuous on Interval|continuous]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$.
Then:
:$\ds \size {\int_a^b \map f t \rd t} \le \int_a^b \size {\map f t} \rd t$ | From [[Negative of Absolute Value]], we have for all $a \in \closedint a b$:
:$-\size {\map f t} \le \map f t \le \size {\map f t}$
Thus from [[Relative Sizes of Definite Integrals]]:
:$\ds -\int_a^b \size {\map f t} \rd t \le \int_a^b \map f t \rd t \le \int_a^b \size {\map f t} \rd t$
Hence the result.
{{qed}} | Triangle Inequality for Integrals/Real | https://proofwiki.org/wiki/Triangle_Inequality_for_Integrals/Real | https://proofwiki.org/wiki/Triangle_Inequality_for_Integrals/Real | [
"Triangle Inequality for Integrals",
"Integral Calculus"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Interval",
"Definition:Real Interval/Closed"
] | [
"Negative of Absolute Value",
"Relative Sizes of Definite Integrals"
] |
proofwiki-1656 | Cauchy Integral Test | Let $f$ be a real function which is continuous, positive and decreasing on the interval $\hointr 1 {+\infty}$.
Let the sequence $\sequence {\Delta_n}$ be defined as:
:$\ds \Delta_n = \sum_{k \mathop = 1}^n \map f k - \int_1^n \map f x \rd x$
Then $\sequence {\Delta_n} $ is decreasing and bounded below by zero.
Hence it... | From Darboux's Theorem, we have that:
:$\ds m \paren {b - a} \le \int_a^b \map f x \rd x \le M \paren {b - a}$
where:
:$M$ is the maximum
and:
:$m$ is the minimum
of $\map f x$ on $\closedint a b$.
Since $f$ decreases, $M = \map f a$ and $m = \map f b$.
Thus it follows that:
:$\ds \forall k \in \N_{>0}: \map f {k + 1} ... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Continuous Real Function on Half Open Interval|continuous]], [[Definition:Positive Real Function|positive]] and [[Definition:Decreasing Real Function|decreasing]] on the [[Definition:Half-Open Real Interval|interval]] $\hointr 1 {+\infty}$.
... | From [[Darboux's Theorem]], we have that:
:$\ds m \paren {b - a} \le \int_a^b \map f x \rd x \le M \paren {b - a}$
where:
:$M$ is the [[Definition:Maximum Value|maximum]]
and:
:$m$ is the [[Definition:Minimum Value|minimum]]
of $\map f x$ on $\closedint a b$.
Since $f$ decreases, $M = \map f a$ and $m = \map f b$.
Th... | Cauchy Integral Test | https://proofwiki.org/wiki/Cauchy_Integral_Test | https://proofwiki.org/wiki/Cauchy_Integral_Test | [
"Cauchy Integral Test",
"Integral Calculus",
"Convergence Tests",
"Series"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Half Open Interval",
"Definition:Positive Real Function",
"Definition:Decreasing/Real Function",
"Definition:Real Interval/Half-Open",
"Definition:Sequence",
"Definition:Decreasing/Sequence/Real Sequence",
"Definition:Bounded Below Set",... | [
"Darboux's Theorem",
"Definition:Maximum Value of Real Function/Absolute",
"Definition:Minimum Value of Real Function/Absolute",
"Definition:Decreasing/Real Function"
] |
proofwiki-1657 | Harmonic Series is Divergent | The harmonic series:
:$\ds \sum_{n \mathop = 1}^\infty \frac 1 n$
diverges. | :$\ds \sum_{n \mathop = 1}^\infty \frac 1 n = \underbrace 1_{s_0} + \underbrace {\frac 1 2 + \frac 1 3}_{s_1} + \underbrace {\frac 1 4 + \frac 1 5 + \frac 1 6 + \frac 1 7}_{s_2} + \cdots$
where $\ds s_k = \sum_{i \mathop = 2^k}^{2^{k + 1} \mathop - 1} \frac 1 i$
From Ordering of Reciprocals:
:$\forall m, n \in \N_{>0}... | The [[Definition:Harmonic Series|harmonic series]]:
:$\ds \sum_{n \mathop = 1}^\infty \frac 1 n$
[[Definition:Divergent Series|diverges]]. | :$\ds \sum_{n \mathop = 1}^\infty \frac 1 n = \underbrace 1_{s_0} + \underbrace {\frac 1 2 + \frac 1 3}_{s_1} + \underbrace {\frac 1 4 + \frac 1 5 + \frac 1 6 + \frac 1 7}_{s_2} + \cdots$
where $\ds s_k = \sum_{i \mathop = 2^k}^{2^{k + 1} \mathop - 1} \frac 1 i$
From [[Ordering of Reciprocals]]:
:$\forall m, n \in ... | Harmonic Series is Divergent/Proof 1 | https://proofwiki.org/wiki/Harmonic_Series_is_Divergent | https://proofwiki.org/wiki/Harmonic_Series_is_Divergent/Proof_1 | [
"Harmonic Series is Divergent",
"Real Analysis",
"Harmonic Series",
"Examples of Divergent Series"
] | [
"Definition:Harmonic Series",
"Definition:Divergent Series"
] | [
"Ordering of Reciprocals",
"Divergence Test",
"Comparison Test for Divergence"
] |
proofwiki-1658 | Harmonic Series is Divergent | The harmonic series:
:$\ds \sum_{n \mathop = 1}^\infty \frac 1 n$
diverges. | Observe that all the terms of the harmonic series are strictly positive.
From Reciprocal Sequence is Strictly Decreasing, the terms are decreasing.
Hence the Cauchy Condensation Test can be applied, and we examine the convergence of:
{{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 1}^\infty 2^n \frac 1 {2^n}
| r = \s... | The [[Definition:Harmonic Series|harmonic series]]:
:$\ds \sum_{n \mathop = 1}^\infty \frac 1 n$
[[Definition:Divergent Series|diverges]]. | Observe that all the terms of the harmonic series are [[Definition:Strictly Positive|strictly positive]].
From [[Reciprocal Sequence is Strictly Decreasing]], the terms are [[Definition:Decreasing Sequence|decreasing]].
Hence the [[Cauchy Condensation Test]] can be applied, and we examine the convergence of:
{{begin... | Harmonic Series is Divergent/Proof 2 | https://proofwiki.org/wiki/Harmonic_Series_is_Divergent | https://proofwiki.org/wiki/Harmonic_Series_is_Divergent/Proof_2 | [
"Harmonic Series is Divergent",
"Real Analysis",
"Harmonic Series",
"Examples of Divergent Series"
] | [
"Definition:Harmonic Series",
"Definition:Divergent Series"
] | [
"Definition:Strictly Positive",
"Reciprocal Sequence is Strictly Decreasing",
"Definition:Decreasing/Sequence",
"Cauchy Condensation Test",
"Divergence Test"
] |
proofwiki-1659 | Harmonic Series is Divergent | The harmonic series:
:$\ds \sum_{n \mathop = 1}^\infty \frac 1 n$
diverges. | We have that the Integral of Reciprocal is Divergent.
Hence from the Cauchy Integral Test, the harmonic series also diverges.
{{qed}} | The [[Definition:Harmonic Series|harmonic series]]:
:$\ds \sum_{n \mathop = 1}^\infty \frac 1 n$
[[Definition:Divergent Series|diverges]]. | We have that the [[Integral of Reciprocal is Divergent]].
Hence from the [[Cauchy Integral Test]], the [[Definition:Harmonic Series|harmonic series]] also [[Definition:Divergent Series|diverges]].
{{qed}} | Harmonic Series is Divergent/Proof 3 | https://proofwiki.org/wiki/Harmonic_Series_is_Divergent | https://proofwiki.org/wiki/Harmonic_Series_is_Divergent/Proof_3 | [
"Harmonic Series is Divergent",
"Real Analysis",
"Harmonic Series",
"Examples of Divergent Series"
] | [
"Definition:Harmonic Series",
"Definition:Divergent Series"
] | [
"Integral of Reciprocal is Divergent",
"Cauchy Integral Test",
"Definition:Harmonic Series",
"Definition:Divergent Series"
] |
proofwiki-1660 | Harmonic Series is Divergent | The harmonic series:
:$\ds \sum_{n \mathop = 1}^\infty \frac 1 n$
diverges. | For all $N \in \N$:
:$\dfrac 1 N + \dfrac 1 {N + 1} + \cdots + \dfrac 1 {2 N} > N \cdot \dfrac 1 {2 N} = \dfrac 1 2$
Hence, by Cauchy's Convergence Criterion for Series, the Harmonic series is divergent.
{{qed}} | The [[Definition:Harmonic Series|harmonic series]]:
:$\ds \sum_{n \mathop = 1}^\infty \frac 1 n$
[[Definition:Divergent Series|diverges]]. | For all $N \in \N$:
:$\dfrac 1 N + \dfrac 1 {N + 1} + \cdots + \dfrac 1 {2 N} > N \cdot \dfrac 1 {2 N} = \dfrac 1 2$
Hence, by [[Cauchy's Convergence Criterion for Series]], the Harmonic series is divergent.
{{qed}} | Harmonic Series is Divergent/Proof 4 | https://proofwiki.org/wiki/Harmonic_Series_is_Divergent | https://proofwiki.org/wiki/Harmonic_Series_is_Divergent/Proof_4 | [
"Harmonic Series is Divergent",
"Real Analysis",
"Harmonic Series",
"Examples of Divergent Series"
] | [
"Definition:Harmonic Series",
"Definition:Divergent Series"
] | [
"Cauchy's Convergence Criterion for Series"
] |
proofwiki-1661 | Sum of Reciprocals of Primes is Divergent | Let $n \in \N: n \ge 1$.
There exists a (strictly) positive real number $C \in \R_{>0}$ such that:
:$(1): \quad \ds \sum_{\substack {p \mathop \in \Bbb P \\ p \mathop \le n} } \frac 1 p > \map \ln {\ln n} - C$
where $\Bbb P$ is the set of all prime numbers.
:$(2): \quad \ds \lim_{n \mathop \to \infty} \paren {\map \ln ... | By Sum of Reciprocals of Primes is Divergent: Lemma:
:$\ds \lim_{n \mathop \to \infty} \paren {\map \ln {\map \ln n} - \frac 1 2} = +\infty$
{{qed|lemma}}
It remains to be proved that:
:$\ds \sum_{\substack {p \mathop \in \Bbb P \\ p \mathop \le n} } \frac 1 p > \map \ln {\ln n} - \frac 1 2$
Assume all sums and product... | Let $n \in \N: n \ge 1$.
There exists a [[Definition:Strictly Positive Real Number|(strictly) positive real number]] $C \in \R_{>0}$ such that:
:$(1): \quad \ds \sum_{\substack {p \mathop \in \Bbb P \\ p \mathop \le n} } \frac 1 p > \map \ln {\ln n} - C$
where $\Bbb P$ is the [[Definition:Set|set]] of all [[Definit... | By [[Sum of Reciprocals of Primes is Divergent/Lemma|Sum of Reciprocals of Primes is Divergent: Lemma]]:
:$\ds \lim_{n \mathop \to \infty} \paren {\map \ln {\map \ln n} - \frac 1 2} = +\infty$
{{qed|lemma}}
It remains to be proved that:
:$\ds \sum_{\substack {p \mathop \in \Bbb P \\ p \mathop \le n} } \frac 1 p > \ma... | Sum of Reciprocals of Primes is Divergent/Proof 1 | https://proofwiki.org/wiki/Sum_of_Reciprocals_of_Primes_is_Divergent | https://proofwiki.org/wiki/Sum_of_Reciprocals_of_Primes_is_Divergent/Proof_1 | [
"Sum of Reciprocals of Primes is Divergent",
"Examples of Divergent Series",
"Analytic Number Theory"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Set",
"Definition:Prime Number"
] | [
"Sum of Reciprocals of Primes is Divergent/Lemma",
"Definition:Prime Number",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Sum of Infinite Geometric Sequence",
"Fundamental Theorem of Arithmetic",
"Cauchy Integral Test",
"Sum of Logarithms",
"Logarithm of Power/Natural Logari... |
proofwiki-1662 | Sum of Reciprocals of Primes is Divergent | Let $n \in \N: n \ge 1$.
There exists a (strictly) positive real number $C \in \R_{>0}$ such that:
:$(1): \quad \ds \sum_{\substack {p \mathop \in \Bbb P \\ p \mathop \le n} } \frac 1 p > \map \ln {\ln n} - C$
where $\Bbb P$ is the set of all prime numbers.
:$(2): \quad \ds \lim_{n \mathop \to \infty} \paren {\map \ln ... | Let $n \in \N$ be a natural number.
Let $p_n$ denote the $n$th prime number.
Consider the continued product:
:$\ds \prod_{k \mathop = 1}^n \frac 1 {1 - 1 / p_k}$
By Sum of Infinite Geometric Sequence:
{{begin-eqn}}
{{eqn | l = \frac 1 {1 - \frac 1 2}
| r = 1 + \frac 1 2 + \frac 1 {2^2} + \cdots
| c =
}}
{{... | Let $n \in \N: n \ge 1$.
There exists a [[Definition:Strictly Positive Real Number|(strictly) positive real number]] $C \in \R_{>0}$ such that:
:$(1): \quad \ds \sum_{\substack {p \mathop \in \Bbb P \\ p \mathop \le n} } \frac 1 p > \map \ln {\ln n} - C$
where $\Bbb P$ is the [[Definition:Set|set]] of all [[Definit... | Let $n \in \N$ be a [[Definition:Natural Number|natural number]].
Let $p_n$ denote the $n$th [[Definition:Prime Number|prime number]].
Consider the [[Definition:Continued Product|continued product]]:
:$\ds \prod_{k \mathop = 1}^n \frac 1 {1 - 1 / p_k}$
By [[Sum of Infinite Geometric Sequence]]:
{{begin-eqn}}
{{eqn ... | Sum of Reciprocals of Primes is Divergent/Proof 2 | https://proofwiki.org/wiki/Sum_of_Reciprocals_of_Primes_is_Divergent | https://proofwiki.org/wiki/Sum_of_Reciprocals_of_Primes_is_Divergent/Proof_2 | [
"Sum of Reciprocals of Primes is Divergent",
"Examples of Divergent Series",
"Analytic Number Theory"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Set",
"Definition:Prime Number"
] | [
"Definition:Natural Numbers",
"Definition:Prime Number",
"Definition:Continued Product",
"Sum of Infinite Geometric Sequence",
"Definition:Series",
"Definition:Series",
"Definition:Series",
"Definition:Series",
"Definition:Convergent Series",
"Fundamental Theorem of Arithmetic",
"Definition:Inte... |
proofwiki-1663 | Sum of Reciprocals of Primes is Divergent | Let $n \in \N: n \ge 1$.
There exists a (strictly) positive real number $C \in \R_{>0}$ such that:
:$(1): \quad \ds \sum_{\substack {p \mathop \in \Bbb P \\ p \mathop \le n} } \frac 1 p > \map \ln {\ln n} - C$
where $\Bbb P$ is the set of all prime numbers.
:$(2): \quad \ds \lim_{n \mathop \to \infty} \paren {\map \ln ... | {{AimForCont}} the contrary.
If the prime reciprocal series converges then there must exist some $k \in \N$ such that:
:$\ds \sum_{n \mathop = k \mathop + 1}^{\infty} \frac 1 {p_n} < \frac 1 2$
Fixing $k$ from the series above, we now let:
:$\ds Q = \prod_{i \mathop = 1}^k {p_i}$
and:
:$\ds \map S r = \sum_{i \mathop =... | Let $n \in \N: n \ge 1$.
There exists a [[Definition:Strictly Positive Real Number|(strictly) positive real number]] $C \in \R_{>0}$ such that:
:$(1): \quad \ds \sum_{\substack {p \mathop \in \Bbb P \\ p \mathop \le n} } \frac 1 p > \map \ln {\ln n} - C$
where $\Bbb P$ is the [[Definition:Set|set]] of all [[Definit... | {{AimForCont}} the contrary.
If the [[Definition:Prime Number|prime]] [[Definition:Reciprocal|reciprocal]] [[Definition:Series|series]] [[Definition:Convergent Series|converges]] then there must exist some $k \in \N$ such that:
:$\ds \sum_{n \mathop = k \mathop + 1}^{\infty} \frac 1 {p_n} < \frac 1 2$
Fixing $k$ fr... | Sum of Reciprocals of Primes is Divergent/Proof 3 | https://proofwiki.org/wiki/Sum_of_Reciprocals_of_Primes_is_Divergent | https://proofwiki.org/wiki/Sum_of_Reciprocals_of_Primes_is_Divergent/Proof_3 | [
"Sum of Reciprocals of Primes is Divergent",
"Examples of Divergent Series",
"Analytic Number Theory"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Set",
"Definition:Prime Number"
] | [
"Definition:Prime Number",
"Definition:Reciprocal",
"Definition:Series",
"Definition:Convergent Series",
"Definition:Series",
"Definition:Term",
"Definition:Distinct",
"Definition:Prime Number",
"Definition:Coprime/Integers",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Pr... |
proofwiki-1664 | Sum of Reciprocals of Primes is Divergent | Let $n \in \N: n \ge 1$.
There exists a (strictly) positive real number $C \in \R_{>0}$ such that:
:$(1): \quad \ds \sum_{\substack {p \mathop \in \Bbb P \\ p \mathop \le n} } \frac 1 p > \map \ln {\ln n} - C$
where $\Bbb P$ is the set of all prime numbers.
:$(2): \quad \ds \lim_{n \mathop \to \infty} \paren {\map \ln ... | {{AimForCont}} the contrary.
If the prime reciprocal series converges then there must exist some $k \in \N$ such that:
:$\ds \sum_{n \mathop = k + 1}^\infty \frac 1 {p_n} < \frac 1 2$
Fixing $p_k$ from the series above, we create a set of natural numbers called $M_x$.
The set, $M_x$ only contains natural numbers betwee... | Let $n \in \N: n \ge 1$.
There exists a [[Definition:Strictly Positive Real Number|(strictly) positive real number]] $C \in \R_{>0}$ such that:
:$(1): \quad \ds \sum_{\substack {p \mathop \in \Bbb P \\ p \mathop \le n} } \frac 1 p > \map \ln {\ln n} - C$
where $\Bbb P$ is the [[Definition:Set|set]] of all [[Definit... | {{AimForCont}} the contrary.
If the [[Definition:Prime Number|prime]] [[Definition:Reciprocal|reciprocal]] [[Definition:Series|series]] [[Definition:Convergent Series|converges]] then there must exist some $k \in \N$ such that:
:$\ds \sum_{n \mathop = k + 1}^\infty \frac 1 {p_n} < \frac 1 2$
Fixing $p_k$ from the [[... | Sum of Reciprocals of Primes is Divergent/Proof 4 | https://proofwiki.org/wiki/Sum_of_Reciprocals_of_Primes_is_Divergent | https://proofwiki.org/wiki/Sum_of_Reciprocals_of_Primes_is_Divergent/Proof_4 | [
"Sum of Reciprocals of Primes is Divergent",
"Examples of Divergent Series",
"Analytic Number Theory"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Set",
"Definition:Prime Number"
] | [
"Definition:Prime Number",
"Definition:Reciprocal",
"Definition:Series",
"Definition:Convergent Series",
"Definition:Series",
"Definition:Set",
"Definition:Natural Numbers",
"Definition:Set",
"Definition:Natural Numbers",
"Definition:Divisor (Algebra)/Integer",
"Definition:Prime Number",
"Defi... |
proofwiki-1665 | Measure of Interval is Length | Let $I$ be a real interval whose endpoints are $a$ and $b$.
Then $I$ is Lebesgue measurable, and the value of the measure is the length of the interval $b - a$. | {{link wanted|Insert here: Some stuff proving intervals are measurable}}
Let $L \subset \R$ be a real interval.
Then $L$ has two distinct endpoints: $a$ and $b$.
Let $\ds \set {I_n}_{n \mathop = 1}^\infty$ be a set of open real intervals satisfying:
:$\ds L \subseteq \bigcup_{n \mathop = 1}^\infty I_n$ | Let $I$ be a [[Definition:Real Interval|real interval]] whose [[Definition:Endpoint of Real Interval|endpoints]] are $a$ and $b$.
Then $I$ is [[Definition:Measurable Set|Lebesgue measurable]], and the value of the [[Definition:Lebesgue Measure|measure]] is the [[Definition:Length of Real Interval|length of the interva... | {{link wanted|Insert here: Some stuff proving intervals are measurable}}
Let $L \subset \R$ be a [[Definition:Real Interval|real interval]].
Then $L$ has two [[Definition:Distinct Objects|distinct]] [[Definition:Endpoint of Real Interval|endpoints]]: $a$ and $b$.
Let $\ds \set {I_n}_{n \mathop = 1}^\infty$ be a set ... | Measure of Interval is Length | https://proofwiki.org/wiki/Measure_of_Interval_is_Length | https://proofwiki.org/wiki/Measure_of_Interval_is_Length | [
"Lebesgue Measure"
] | [
"Definition:Real Interval",
"Definition:Real Interval/Endpoints",
"Definition:Measurable Set",
"Definition:Lebesgue Measure",
"Definition:Real Interval/Length"
] | [
"Definition:Real Interval",
"Definition:Distinct/Plural",
"Definition:Real Interval/Endpoints",
"Definition:Real Interval/Open",
"Definition:Real Interval/Open",
"Definition:Real Interval/Open",
"Definition:Real Interval/Open"
] |
proofwiki-1666 | Measurable Sets form Algebra of Sets | Let $\mu^*$ be an outer measure on a set $X$.
Then the set of $\mu^*$-measurable sets is an algebra of sets. | For a subset $S \subseteq X$, let $\relcomp X S$ denote the relative complement of $S$ in $X$.
We first prove the second property of an algebra of sets, as described on that page.
Let $S$ be $\mu^*$-measurable. For any subset $A \subseteq X$:
{{begin-eqn}}
{{eqn | l = \map {\mu^*} A
| r = \map {\mu^*} {A \cap S} ... | Let $\mu^*$ be an [[Definition:Outer Measure|outer measure]] on a [[Definition:Set|set]] $X$.
Then the set of [[Definition:Measurable Set#Measurable Sets of an Arbitrary Outer Measure|$\mu^*$-measurable sets]] is an [[Definition:Algebra of Sets|algebra of sets]]. | For a [[Definition:Subset|subset]] $S \subseteq X$, let $\relcomp X S$ denote the [[Definition:Relative Complement|relative complement]] of $S$ in $X$.
We first prove the second property of an [[Definition:Algebra of Sets|algebra of sets]], as described on that page.
Let $S$ be $\mu^*$-measurable. For any [[Definitio... | Measurable Sets form Algebra of Sets | https://proofwiki.org/wiki/Measurable_Sets_form_Algebra_of_Sets | https://proofwiki.org/wiki/Measurable_Sets_form_Algebra_of_Sets | [
"Measurable Sets",
"Algebras of Sets"
] | [
"Definition:Outer Measure",
"Definition:Set",
"Definition:Measurable Set",
"Definition:Algebra of Sets"
] | [
"Definition:Subset",
"Definition:Relative Complement",
"Definition:Algebra of Sets",
"Definition:Subset",
"Complement of Complement",
"Definition:Subset",
"Set Difference Union Intersection",
"Union is Commutative",
"Intersection is Associative",
"Intersection is Commutative",
"Intersection with... |
proofwiki-1667 | Classification of Compact Three-Manifolds Supporting Zero-Curvature Geometry | Every closed, orientable, path connected $3$-dimensional Riemannian manifold which supports a geometry of zero curvature is homeomorphic to one of the following:
* Torus $\mathbb T^3$
* Half-Twist Cube
* Quarter-Twist Cube
* Hantschze-Wendt Manifold
* $\frac 1 6$-Twist Hexagonal Prism
* $\frac 1 3$-Twist Hexagonal Pris... | {{ProofWanted}}
Category:Riemannian Manifolds
2z2tl8jqvzmy7cab8r58hk6yezorfuf | Every [[Definition:Closed Set (Topology)|closed]], [[Definition:Orientable|orientable]], [[Definition:Path-Connected|path connected]] $3$-[[Definition:Dimension (Topology)|dimensional]] [[Definition:Riemannian Manifold|Riemannian manifold]] which supports a [[Definition:Riemannian Metric|geometry]] of [[Definition:Zero... | {{ProofWanted}}
[[Category:Riemannian Manifolds]]
2z2tl8jqvzmy7cab8r58hk6yezorfuf | Classification of Compact Three-Manifolds Supporting Zero-Curvature Geometry | https://proofwiki.org/wiki/Classification_of_Compact_Three-Manifolds_Supporting_Zero-Curvature_Geometry | https://proofwiki.org/wiki/Classification_of_Compact_Three-Manifolds_Supporting_Zero-Curvature_Geometry | [
"Riemannian Manifolds"
] | [
"Definition:Closed Set/Topology",
"Definition:Orientable",
"Definition:Path-Connected",
"Definition:Dimension (Topology)",
"Definition:Riemannian Manifold",
"Definition:Riemannian Metric",
"Definition:Zero Gaussian Curvature",
"Definition:Gaussian Curvature",
"Definition:Homeomorphism",
"Definitio... | [
"Category:Riemannian Manifolds"
] |
proofwiki-1668 | Heine-Borel Theorem/Real Line | Let $\R$ be the real number line considered as a (real) Euclidean space.
Let $C \subseteq \R$.
Then $C$ is closed and bounded in $\R$ {{iff}} $C$ is compact. | === Necessary Condition ===
Let $C$ be closed and bounded in $\R$.
Then, by Closed Bounded Subset of Real Numbers is Compact, $C$ is compact.
{{qed|lemma}} | Let $\R$ be the [[Definition:Real Number Line|real number line]] considered as a [[Definition:Real Euclidean Space|(real) Euclidean space]].
Let $C \subseteq \R$.
Then $C$ is [[Definition:Closed Set (Metric Space)|closed]] and [[Definition:Bounded Metric Space|bounded]] in $\R$ {{iff}} $C$ is [[Definition:Compact Su... | === Necessary Condition ===
Let $C$ be [[Definition:Closed Set (Metric Space)|closed]] and [[Definition:Bounded Metric Space|bounded]] in $\R$.
Then, by [[Closed Bounded Subset of Real Numbers is Compact]], $C$ is [[Definition:Compact Subspace|compact]].
{{qed|lemma}} | Heine-Borel Theorem/Real Line | https://proofwiki.org/wiki/Heine-Borel_Theorem/Real_Line | https://proofwiki.org/wiki/Heine-Borel_Theorem/Real_Line | [
"Heine-Borel Theorem",
"Real Euclidean Spaces",
"Compact Spaces (Real Analysis)",
"Real Analysis"
] | [
"Definition:Real Number/Real Number Line",
"Definition:Euclidean Space/Real",
"Definition:Closed Set/Metric Space",
"Definition:Bounded Metric Space",
"Definition:Compact Topological Space/Subspace"
] | [
"Definition:Closed Set/Metric Space",
"Definition:Bounded Metric Space",
"Closed Bounded Subset of Real Numbers is Compact",
"Definition:Compact Topological Space/Subspace",
"Definition:Compact Topological Space/Subspace",
"Definition:Closed Set/Metric Space",
"Definition:Bounded Metric Space"
] |
proofwiki-1669 | Measurable Image | Let $\mathfrak M$ be the set of measurable sets of $\R$.
For any extended real-valued function $f: \R \to \R \cup \set {-\infty, +\infty}$ whose domain is measurable, the following statements are equivalent:
:$(1): \quad \forall \alpha \in \R: \set {x: \map f x > \alpha} \in \mathfrak M$
:$(2): \quad \forall \alpha \in... | Let the domain of $f$ be $D$.
We have that Measurable Sets form Algebra of Sets.
First we note that, from Properties of Algebras of Sets, the difference of two measurable sets is measurable.
So:
:$\set {x: \map f x \le \alpha} = D - \set {x: \map f x > \alpha}$
and so $(1) \iff (4)$.
Similarly, $(2) \iff (3)$.
Next we ... | Let $\mathfrak M$ be the set of [[Definition:Measurable Set|measurable sets]] of $\R$.
For any [[Definition:Extended Real-Valued Function|extended real-valued function]] $f: \R \to \R \cup \set {-\infty, +\infty}$ whose [[Definition:Domain of Mapping|domain]] is [[Definition:Measurable Set|measurable]], the following ... | Let the [[Definition:Domain of Mapping|domain]] of $f$ be $D$.
We have that [[Measurable Sets form Algebra of Sets]].
First we note that, from [[Properties of Algebras of Sets]], the [[Definition:Set Difference|difference]] of two [[Definition:Measurable Set|measurable sets]] is measurable.
So:
:$\set {x: \map f x ... | Measurable Image | https://proofwiki.org/wiki/Measurable_Image | https://proofwiki.org/wiki/Measurable_Image | [
"Measure Theory"
] | [
"Definition:Measurable Set",
"Definition:Extended Real-Valued Function",
"Definition:Domain (Set Theory)/Mapping",
"Definition:Measurable Set"
] | [
"Definition:Domain (Set Theory)/Mapping",
"Measurable Sets form Algebra of Sets",
"Properties of Algebras of Sets",
"Definition:Set Difference",
"Definition:Measurable Set",
"Properties of Algebras of Sets",
"Definition:Set Intersection",
"Definition:Sequence",
"Definition:Measurable Set",
"Catego... |
proofwiki-1670 | Lebesgue Integral is Extension of Darboux Integral | Let $f: \closedint a b \to \R$ be a Darboux integrable function.
Then it is also Lebesgue integrable, and furthermore:
:$\ds R \int_a^b \map f x \rd x = \int_{\closedint a b} f \rd \lambda$
where $\ds R \int_a^b$ is the Darboux integral and $\ds \int_{\closedint a b}$ is the Lebesgue integral. | Since every step function is also a simple function, we have
:$\ds \map L P \le \sup_{\phi \mathop \le f} \int_a^b \map \phi x \rd x \le \inf_{\psi \mathop \ge f} \int_a^b \map \psi x \rd x \le \map U P$
where $\map L P$ and $\map U P$ are the lower Darboux sum and upper Darboux sum as defined in the definition of defi... | Let $f: \closedint a b \to \R$ be a [[Definition:Darboux Integrable Function|Darboux integrable function]].
Then it is also [[Definition:Lebesgue Integrable Function|Lebesgue integrable]], and furthermore:
:$\ds R \int_a^b \map f x \rd x = \int_{\closedint a b} f \rd \lambda$
where $\ds R \int_a^b$ is the [[Definiti... | Since every [[Definition:Step Function|step function]] is also a [[Definition:Simple Function|simple function]], we have
:$\ds \map L P \le \sup_{\phi \mathop \le f} \int_a^b \map \phi x \rd x \le \inf_{\psi \mathop \ge f} \int_a^b \map \psi x \rd x \le \map U P$
where $\map L P$ and $\map U P$ are the [[Definition:L... | Lebesgue Integral is Extension of Darboux Integral | https://proofwiki.org/wiki/Lebesgue_Integral_is_Extension_of_Darboux_Integral | https://proofwiki.org/wiki/Lebesgue_Integral_is_Extension_of_Darboux_Integral | [
"Analysis",
"Integral Calculus",
"Measure Theory"
] | [
"Definition:Darboux Integrable Function",
"Definition:Integrable Function/Lebesgue",
"Definition:Definite Integral/Darboux",
"Definition:Lebesgue Integral"
] | [
"Definition:Step Function",
"Definition:Simple Function",
"Definition:Lower Darboux Sum",
"Definition:Upper Darboux Sum",
"Definition:Definite Integral",
"Definition:Darboux Integrable Function",
"Infimum and Supremum Equivalent for Measurable Function"
] |
proofwiki-1671 | Euclidean Metric on Real Vector Space is Metric | The Euclidean metric on a real vector space $\R^n$ is a metric. | The Euclidean metric on $\R^n$ is a special case of the $p$-product metric.
The result follows from $p$-Product Metric on Real Vector Space is Metric.
{{qed}} | The [[Definition:Euclidean Metric on Real Vector Space|Euclidean metric]] on a [[Definition:Real Vector Space|real vector space]] $\R^n$ is a [[Definition:Metric|metric]]. | The [[Definition:Euclidean Metric on Real Vector Space|Euclidean metric]] on $\R^n$ is a special case of the [[Definition:P-Product Metric on Real Vector Space|$p$-product metric]].
The result follows from [[P-Product Metric on Real Vector Space is Metric|$p$-Product Metric on Real Vector Space is Metric]].
{{qed}} | Euclidean Metric on Real Vector Space is Metric/Proof 1 | https://proofwiki.org/wiki/Euclidean_Metric_on_Real_Vector_Space_is_Metric | https://proofwiki.org/wiki/Euclidean_Metric_on_Real_Vector_Space_is_Metric/Proof_1 | [
"Euclidean Metric",
"Euclidean Metric on Real Vector Space is Metric"
] | [
"Definition:Euclidean Metric/Real Vector Space",
"Definition:Real Vector Space",
"Definition:Metric Space/Metric"
] | [
"Definition:Euclidean Metric/Real Vector Space",
"Definition:P-Product Metric/Real Vector Space",
"P-Product Metric on Real Vector Space is Metric"
] |
proofwiki-1672 | Euclidean Metric on Real Vector Space is Metric | The Euclidean metric on a real vector space $\R^n$ is a metric. | Consider the Euclidean space $M = \struct {\R^n, d_2}$ where $d_2$ is the distance function given by:
:$\ds \map {d_2} {x, y} = \paren {\sum_{i \mathop = 1}^n \paren {x_i - y_i}^2}^{\frac 1 2}$
where $x = \tuple {x_1, x_2, \ldots, x_n}$ and $y = \tuple {y_1, y_2, \ldots, y_n}$.
=== Proof of {{Metric-space-axiom|1|nolin... | The [[Definition:Euclidean Metric on Real Vector Space|Euclidean metric]] on a [[Definition:Real Vector Space|real vector space]] $\R^n$ is a [[Definition:Metric|metric]]. | Consider the [[Definition:Real Euclidean Space|Euclidean space]] $M = \struct {\R^n, d_2}$ where $d_2$ is the [[Definition:Distance Function|distance function]] given by:
:$\ds \map {d_2} {x, y} = \paren {\sum_{i \mathop = 1}^n \paren {x_i - y_i}^2}^{\frac 1 2}$
where $x = \tuple {x_1, x_2, \ldots, x_n}$ and $y = \tupl... | Euclidean Metric on Real Vector Space is Metric/Proof 2 | https://proofwiki.org/wiki/Euclidean_Metric_on_Real_Vector_Space_is_Metric | https://proofwiki.org/wiki/Euclidean_Metric_on_Real_Vector_Space_is_Metric/Proof_2 | [
"Euclidean Metric",
"Euclidean Metric on Real Vector Space is Metric"
] | [
"Definition:Euclidean Metric/Real Vector Space",
"Definition:Real Vector Space",
"Definition:Metric Space/Metric"
] | [
"Definition:Euclidean Space/Real",
"Definition:Distance Function",
"Minkowski's Inequality for Sums/Index 2"
] |
proofwiki-1673 | Metric Induces Topology | Let $M = \struct {A, d}$ be a metric space.
Then the topology $\tau$ induced by the metric $d$ is a topology on $M$. | We examine each of the criteria for being a topology separately.
:$(1): \quad$ By Union of Open Sets of Metric Space is Open, the union of any collection of open sets of a metric space is open.
:$(2): \quad$ By Finite Intersection of Open Sets of Metric Space is Open, a finite intersection of open sets of a metric spac... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Then the [[Definition:Topology Induced by Metric|topology $\tau$ induced]] by the [[Definition:Metric|metric]] $d$ is a [[Definition:Topology|topology]] on $M$. | We examine each of the criteria for being a [[Definition:Topology|topology]] separately.
:$(1): \quad$ By [[Union of Open Sets of Metric Space is Open]], the [[Definition:Set Union|union]] of any collection of [[Definition:Open Set of Metric Space|open sets]] of a [[Definition:Metric Space|metric space]] is [[Definiti... | Metric Induces Topology | https://proofwiki.org/wiki/Metric_Induces_Topology | https://proofwiki.org/wiki/Metric_Induces_Topology | [
"Metric Induces Topology",
"Topologies Induced by Metrics"
] | [
"Definition:Metric Space",
"Definition:Topology Induced by Metric",
"Definition:Metric Space/Metric",
"Definition:Topology"
] | [
"Definition:Topology",
"Union of Open Sets of Metric Space is Open",
"Definition:Set Union",
"Definition:Open Set/Metric Space",
"Definition:Metric Space",
"Definition:Open Set/Metric Space",
"Finite Intersection of Open Sets of Metric Space is Open",
"Definition:Set Intersection/Finite Intersection",... |
proofwiki-1674 | Countable Set has Measure Zero | Let $S$ be a countable set.
{{explain|Is it assumed that $S \subseteq \R$?}}
Then the measure of $S$ is $\map m S = 0$. | Let $\ds \set {x_i}_{i \mathop = 1}^\infty$ be an enumeration of the elements of $S$.
For any (strictly) positive real number $\epsilon$, define $A_i$ as an open cover for $S$:
:$A_i = \paren {x_i - 2^{-i} \epsilon, x_i + 2^{-i} \epsilon}$
{{explain|What is the meaning of the above? Is it an open interval? If so, use t... | Let $S$ be a [[Definition:Countable Set|countable set]].
{{explain|Is it assumed that $S \subseteq \R$?}}
Then the [[Definition:Lebesgue Measure|measure]] of $S$ is $\map m S = 0$. | Let $\ds \set {x_i}_{i \mathop = 1}^\infty$ be an enumeration of the [[Definition:Element|elements]] of $S$.
For any [[Definition:Strictly Positive Real Number|(strictly) positive real number]] $\epsilon$, define $A_i$ as an [[Definition:Open Cover|open cover]] for $S$:
:$A_i = \paren {x_i - 2^{-i} \epsilon, x_i + 2^... | Countable Set has Measure Zero | https://proofwiki.org/wiki/Countable_Set_has_Measure_Zero | https://proofwiki.org/wiki/Countable_Set_has_Measure_Zero | [
"Analysis"
] | [
"Definition:Countable Set",
"Definition:Lebesgue Measure"
] | [
"Definition:Element",
"Definition:Strictly Positive/Real Number",
"Definition:Open Cover",
"Sum of Infinite Geometric Sequence",
"Definition:Arbitrary Constant",
"Definition:Strictly Positive/Real Number",
"Definition:Set",
"Definition:Null Set",
"Category:Analysis"
] |
proofwiki-1675 | Mean Value of Convex Real Function | Let $f$ be a real function which is continuous on the closed interval $\closedint a b$ and differentiable on the open interval $\openint a b$.
Let $f$ be convex on $\openint a b$.
Then:
:$\forall \xi \in \openint a b: \map f x - \map f \xi \ge \map {f'} \xi \paren {x - \xi}$ | By the Mean Value Theorem:
:$\exists \eta \in \openint x \xi: \map {f'} \eta = \dfrac {\map f x - \map f \xi} {x - \xi}$
From Real Function is Convex iff Derivative is Increasing, the derivative of $f$ is increasing.
Thus:
:$x > \xi \implies \map {f'} \eta \ge \map {f'} \xi$
:$x < \xi \implies \map {f'} \eta \le \map {... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Continuous on Interval|continuous]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$ and [[Definition:Differentiable on Interval|differentiable]] on the [[Definition:Open Real Interval|open interval]] $\openint a b$... | By the [[Mean Value Theorem]]:
:$\exists \eta \in \openint x \xi: \map {f'} \eta = \dfrac {\map f x - \map f \xi} {x - \xi}$
From [[Real Function is Convex iff Derivative is Increasing]], the [[Definition:Derivative|derivative]] of $f$ is [[Definition:Increasing Real Function|increasing]].
Thus:
:$x > \xi \implies \m... | Mean Value of Convex Real Function | https://proofwiki.org/wiki/Mean_Value_of_Convex_Real_Function | https://proofwiki.org/wiki/Mean_Value_of_Convex_Real_Function | [
"Convex Real Functions",
"Differential Calculus"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Interval",
"Definition:Real Interval/Closed",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Real Interval/Open",
"Definition:Convex Real Function"
] | [
"Mean Value Theorem",
"Real Function is Convex iff Derivative is Increasing",
"Definition:Derivative",
"Definition:Increasing/Real Function"
] |
proofwiki-1676 | Upper Bound of Natural Logarithm | Let $\ln x$ be the natural logarithm of $x$ where $x \in \R_{>0}$.
Then:
:$\ln x \le x - 1$ | From Logarithm is Strictly Concave:
:$\ln$ is (strictly) concave.
From Mean Value of Concave Real Function:
:$\ln x - \ln 1 \le \paren {\dfrac \d {\d x} \ln 1} \paren {x - 1}$
From Derivative of Natural Logarithm:
:$\dfrac \d {\d x} \ln 1 = \dfrac 1 1 = 1$
So:
:$\ln x - \ln 1 \le \paren {x - 1}$
But from Logarithm of 1... | Let $\ln x$ be the [[Definition:Natural Logarithm|natural logarithm]] of $x$ where $x \in \R_{>0}$.
Then:
:$\ln x \le x - 1$ | From [[Logarithm is Strictly Concave]]:
:$\ln$ is [[Definition:Strictly Concave Real Function|(strictly) concave]].
From [[Mean Value of Concave Real Function]]:
:$\ln x - \ln 1 \le \paren {\dfrac \d {\d x} \ln 1} \paren {x - 1}$
From [[Derivative of Natural Logarithm]]:
:$\dfrac \d {\d x} \ln 1 = \dfrac 1 1 = 1$
So... | Upper Bound of Natural Logarithm/Proof 1 | https://proofwiki.org/wiki/Upper_Bound_of_Natural_Logarithm | https://proofwiki.org/wiki/Upper_Bound_of_Natural_Logarithm/Proof_1 | [
"Natural Logarithms",
"Inequalities",
"Upper Bound of Natural Logarithm"
] | [
"Definition:Natural Logarithm"
] | [
"Logarithm is Strictly Concave",
"Definition:Strictly Concave Real Function",
"Mean Value of Concave Real Function",
"Derivative of Natural Logarithm Function",
"Natural Logarithm of 1 is 0"
] |
proofwiki-1677 | Upper Bound of Natural Logarithm | Let $\ln x$ be the natural logarithm of $x$ where $x \in \R_{>0}$.
Then:
:$\ln x \le x - 1$ | Let $\sequence {f_n}$ denote the sequence of mappings $f_n: \R_{>0} \to \R$ defined as:
:$\map {f_n} x = n \paren {\sqrt [n] x - 1}$
Fix $x \in \R_{>0}$.
We first show that
$\forall n \in \N : n \paren {\sqrt [n] x - 1} < x - 1 $
=== Case 1: $0 < x < 1$ ===
Suppose $0 < x < 1$.
Then:
{{begin-eqn}}
{{eqn | l = 0
|... | Let $\ln x$ be the [[Definition:Natural Logarithm|natural logarithm]] of $x$ where $x \in \R_{>0}$.
Then:
:$\ln x \le x - 1$ | Let $\sequence {f_n}$ denote the [[Definition:Sequence|sequence]] of mappings $f_n: \R_{>0} \to \R$ defined as:
:$\map {f_n} x = n \paren {\sqrt [n] x - 1}$
Fix $x \in \R_{>0}$.
We first show that
$\forall n \in \N : n \paren {\sqrt [n] x - 1} < x - 1 $
=== Case 1: $0 < x < 1$ ===
Suppose $0 < x < 1$.
Then:
{{beg... | Upper Bound of Natural Logarithm/Proof 2 | https://proofwiki.org/wiki/Upper_Bound_of_Natural_Logarithm | https://proofwiki.org/wiki/Upper_Bound_of_Natural_Logarithm/Proof_2 | [
"Natural Logarithms",
"Inequalities",
"Upper Bound of Natural Logarithm"
] | [
"Definition:Natural Logarithm"
] | [
"Definition:Sequence",
"Power Function on Base between Zero and One is Strictly Decreasing/Rational Number",
"Real Number Ordering is Compatible with Addition",
"Ordering of Reciprocals",
"Sum of Geometric Sequence",
"Natural Logarithm of 1 is 0/Proof 3",
"Power Function on Base Greater than One is Stri... |
proofwiki-1678 | Exponential of Sum/Real Numbers | Let $x, y \in \R$ be real numbers.
Let $\exp x$ be the exponential of $x$.
Then:
:$\map \exp {x + y} = \paren {\exp x} \paren {\exp y}$ | This proof assumes the definition of $\exp$ as:
:$\exp x = y \iff \ln y = x$
where:
:$\ln y = \ds \int_1^y \dfrac 1 t \rd t$
Let $X = \exp x$ and $Y = \exp y$.
From Sum of Logarithms, we have:
:$\ln X Y = \ln X + \ln Y = x + y$
From the Exponential of Natural Logarithm:
:$\map \exp {\ln x} = x$
Thus:
:$\map \exp {x + y... | Let $x, y \in \R$ be [[Definition:Real Number|real numbers]].
Let $\exp x$ be the [[Definition:Real Exponential Function|exponential of $x$]].
Then:
:$\map \exp {x + y} = \paren {\exp x} \paren {\exp y}$ | This proof assumes the definition of $\exp$ as:
:$\exp x = y \iff \ln y = x$
where:
:$\ln y = \ds \int_1^y \dfrac 1 t \rd t$
Let $X = \exp x$ and $Y = \exp y$.
From [[Sum of Logarithms]], we have:
:$\ln X Y = \ln X + \ln Y = x + y$
From the [[Exponential of Natural Logarithm]]:
:$\map \exp {\ln x} = x$
Thus:
:$... | Exponential of Sum/Real Numbers/Proof 1 | https://proofwiki.org/wiki/Exponential_of_Sum/Real_Numbers | https://proofwiki.org/wiki/Exponential_of_Sum/Real_Numbers/Proof_1 | [
"Exponential of Sum"
] | [
"Definition:Real Number",
"Definition:Exponential Function/Real"
] | [
"Sum of Logarithms",
"Exponential of Natural Logarithm"
] |
proofwiki-1679 | Exponential of Sum/Real Numbers | Let $x, y \in \R$ be real numbers.
Let $\exp x$ be the exponential of $x$.
Then:
:$\map \exp {x + y} = \paren {\exp x} \paren {\exp y}$ | This proof assumes the definition of $\exp$ as defined by a limit of a sequence:
:$\exp x = \ds \lim_{n \mathop \to +\infty} \paren {1 + \frac x n}^n$
From Powers of Group Elements we can presuppose the Exponent Combination Laws for natural number indices.
First we introduce a lemma:
By definition:
{{begin-eqn}}
{{eqn ... | Let $x, y \in \R$ be [[Definition:Real Number|real numbers]].
Let $\exp x$ be the [[Definition:Real Exponential Function|exponential of $x$]].
Then:
:$\map \exp {x + y} = \paren {\exp x} \paren {\exp y}$ | This proof assumes the definition of $\exp$ as defined by a [[Definition:Exponential Function/Real/Limit of Sequence|limit of a sequence]]:
:$\exp x = \ds \lim_{n \mathop \to +\infty} \paren {1 + \frac x n}^n$
From [[Powers of Group Elements]] we can presuppose the [[Exponent Combination Laws]] for [[Definition:Natu... | Exponential of Sum/Real Numbers/Proof 2 | https://proofwiki.org/wiki/Exponential_of_Sum/Real_Numbers | https://proofwiki.org/wiki/Exponential_of_Sum/Real_Numbers/Proof_2 | [
"Exponential of Sum"
] | [
"Definition:Real Number",
"Definition:Exponential Function/Real"
] | [
"Definition:Exponential Function/Real/Limit of Sequence",
"Powers of Group Elements",
"Exponent Combination Laws",
"Definition:Natural Numbers",
"Definition:Power (Algebra)/Exponent",
"Definition:Lemma",
"Combination Theorem for Sequences",
"Exponential of Sum/Real Numbers/Lemma",
"Combination Theor... |
proofwiki-1680 | Exponential of Sum/Real Numbers | Let $x, y \in \R$ be real numbers.
Let $\exp x$ be the exponential of $x$.
Then:
:$\map \exp {x + y} = \paren {\exp x} \paren {\exp y}$ | This proof assumes the definition of $\exp$ as defined by a limit of a sequence:
:$\exp x = \ds \lim_{n \mathop \to +\infty} \paren {1 + \frac x n}^n$
From Powers of Group Elements we can presuppose the Exponent Combination Laws for natural number indices.
By definition:
{{begin-eqn}}
{{eqn | l = \paren {\exp x} \paren... | Let $x, y \in \R$ be [[Definition:Real Number|real numbers]].
Let $\exp x$ be the [[Definition:Real Exponential Function|exponential of $x$]].
Then:
:$\map \exp {x + y} = \paren {\exp x} \paren {\exp y}$ | This proof assumes the definition of $\exp$ as defined by a [[Definition:Exponential Function/Real/Limit of Sequence|limit of a sequence]]:
:$\exp x = \ds \lim_{n \mathop \to +\infty} \paren {1 + \frac x n}^n$
From [[Powers of Group Elements]] we can presuppose the [[Exponent Combination Laws]] for [[Definition:Natu... | Exponential of Sum/Real Numbers/Proof 3 | https://proofwiki.org/wiki/Exponential_of_Sum/Real_Numbers | https://proofwiki.org/wiki/Exponential_of_Sum/Real_Numbers/Proof_3 | [
"Exponential of Sum"
] | [
"Definition:Real Number",
"Definition:Exponential Function/Real"
] | [
"Definition:Exponential Function/Real/Limit of Sequence",
"Powers of Group Elements",
"Exponent Combination Laws",
"Definition:Natural Numbers",
"Definition:Power (Algebra)/Exponent",
"Combination Theorem for Limits of Functions/Real",
"Exponential of Sum/Real Numbers/Lemma",
"Binomial Theorem",
"Co... |
proofwiki-1681 | Exponential of Sum/Real Numbers | Let $x, y \in \R$ be real numbers.
Let $\exp x$ be the exponential of $x$.
Then:
:$\map \exp {x + y} = \paren {\exp x} \paren {\exp y}$ | This proof assumes the definition of $\exp$ as defined by an initial value problem.
That is, suppose $\exp$ satisfies:
:$(1): \quad D_x \exp x = \exp x$
:$(2): \quad \exp 0 = 1$
on $\R$.
Consider the real function $f: \R \to \R$ defined by:
:$\map f x := \dfrac {\map \exp {x + y} } {\map \exp y}$
From Exponential of Re... | Let $x, y \in \R$ be [[Definition:Real Number|real numbers]].
Let $\exp x$ be the [[Definition:Real Exponential Function|exponential of $x$]].
Then:
:$\map \exp {x + y} = \paren {\exp x} \paren {\exp y}$ | This proof assumes the definition of $\exp$ as defined by an [[Definition:Exponential Function/Real/Differential Equation|initial value problem]].
That is, suppose $\exp$ satisfies:
:$(1): \quad D_x \exp x = \exp x$
:$(2): \quad \exp 0 = 1$
on $\R$.
Consider the [[Definition:Real Function|real function]] $f: \R \to ... | Exponential of Sum/Real Numbers/Proof 4 | https://proofwiki.org/wiki/Exponential_of_Sum/Real_Numbers | https://proofwiki.org/wiki/Exponential_of_Sum/Real_Numbers/Proof_4 | [
"Exponential of Sum"
] | [
"Definition:Real Number",
"Definition:Exponential Function/Real"
] | [
"Definition:Exponential Function/Real/Differential Equation",
"Definition:Real Function",
"Exponential of Real Number is Strictly Positive",
"Definition:Well-Defined/Mapping",
"Derivative of Constant Multiple",
"Derivative of Composite Function",
"Exponential Function is Well-Defined/Real/Proof 5"
] |
proofwiki-1682 | Exponential of Sum/Real Numbers | Let $x, y \in \R$ be real numbers.
Let $\exp x$ be the exponential of $x$.
Then:
:$\map \exp {x + y} = \paren {\exp x} \paren {\exp y}$ | This proof assumes the definition of $\exp$ as a series.
Then:
{{begin-eqn}}
{{eqn | l = \map \exp {x + y}
| r = \sum_{n \mathop = 0}^\infty \frac 1 {n!} \paren {x + y}^n
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \frac 1 {n!} \sum_{k \mathop = 0}^n \frac {n!} {k! \paren {n - k}!} x^k y^{n - k}
| c = Binomi... | Let $x, y \in \R$ be [[Definition:Real Number|real numbers]].
Let $\exp x$ be the [[Definition:Real Exponential Function|exponential of $x$]].
Then:
:$\map \exp {x + y} = \paren {\exp x} \paren {\exp y}$ | This proof assumes the definition of $\exp$ [[Definition:Exponential Function/Real/Power Series Expansion|as a series]].
Then:
{{begin-eqn}}
{{eqn | l = \map \exp {x + y}
| r = \sum_{n \mathop = 0}^\infty \frac 1 {n!} \paren {x + y}^n
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \frac 1 {n!} \sum_{k \mathop = 0}^... | Exponential of Sum/Real Numbers/Proof 5 | https://proofwiki.org/wiki/Exponential_of_Sum/Real_Numbers | https://proofwiki.org/wiki/Exponential_of_Sum/Real_Numbers/Proof_5 | [
"Exponential of Sum"
] | [
"Definition:Real Number",
"Definition:Exponential Function/Real"
] | [
"Definition:Exponential Function/Real/Power Series Expansion",
"Binomial Theorem"
] |
proofwiki-1683 | Exponential of Sum/Real Numbers | Let $x, y \in \R$ be real numbers.
Let $\exp x$ be the exponential of $x$.
Then:
:$\map \exp {x + y} = \paren {\exp x} \paren {\exp y}$ | Fix $a \in \R$ and define the function $f_a : \R \to \R$ by:
:$\map {f_a} x = \map \exp {a - x} \exp x$
for all $x \in \R$.
We aim to establish that:
:$\map {f_a} x = \map \exp {a - x} \exp x = \exp a$
for all $a, x \in \R$.
Then, we can fix $x, y \in \R$ and set $a = x + y$ to obtain:
:$\map {f_a} x = \exp y \exp ... | Let $x, y \in \R$ be [[Definition:Real Number|real numbers]].
Let $\exp x$ be the [[Definition:Real Exponential Function|exponential of $x$]].
Then:
:$\map \exp {x + y} = \paren {\exp x} \paren {\exp y}$ | Fix $a \in \R$ and define the [[Definition:Real Function|function]] $f_a : \R \to \R$ by:
:$\map {f_a} x = \map \exp {a - x} \exp x$
for all $x \in \R$.
We aim to establish that:
:$\map {f_a} x = \map \exp {a - x} \exp x = \exp a$
for all $a, x \in \R$.
Then, we can fix $x, y \in \R$ and set $a = x + y$ to obt... | Exponential of Sum/Real Numbers/Proof 6 | https://proofwiki.org/wiki/Exponential_of_Sum/Real_Numbers | https://proofwiki.org/wiki/Exponential_of_Sum/Real_Numbers/Proof_6 | [
"Exponential of Sum"
] | [
"Definition:Real Number",
"Definition:Exponential Function/Real"
] | [
"Definition:Real Function",
"Definition:Differentiable Mapping/Real Function",
"Product Rule for Derivatives",
"Derivative of Composite Function",
"Derivative of Exponential Function",
"Zero Derivative implies Constant Function",
"Definition:Constant Mapping",
"Exponential of Zero"
] |
proofwiki-1684 | Exponential of Sum/Real Numbers | Let $x, y \in \R$ be real numbers.
Let $\exp x$ be the exponential of $x$.
Then:
:$\map \exp {x + y} = \paren {\exp x} \paren {\exp y}$ | Let $\exp$ be defined as the limit of a sequence:
:$\ds \map \exp x = \lim_{n \mathop \to \infty} \paren {1 + \frac x n}^n$
Then:
{{begin-eqn}}
{{eqn | l = \paren {\exp x} \paren {\exp y}
| r = \paren {\lim_{n \mathop \to \infty} \paren {1 + \frac x n}^n} \paren {\lim_{n \mathop \to \infty} \paren {1 + \frac y n}... | Let $x, y \in \R$ be [[Definition:Real Number|real numbers]].
Let $\exp x$ be the [[Definition:Real Exponential Function|exponential of $x$]].
Then:
:$\map \exp {x + y} = \paren {\exp x} \paren {\exp y}$ | Let $\exp$ be defined as the [[Definition:Exponential Function/Real/Limit of Sequence|limit of a sequence]]:
:$\ds \map \exp x = \lim_{n \mathop \to \infty} \paren {1 + \frac x n}^n$
Then:
{{begin-eqn}}
{{eqn | l = \paren {\exp x} \paren {\exp y}
| r = \paren {\lim_{n \mathop \to \infty} \paren {1 + \frac x n}^... | Exponential of Sum/Real Numbers/Proof 7 | https://proofwiki.org/wiki/Exponential_of_Sum/Real_Numbers | https://proofwiki.org/wiki/Exponential_of_Sum/Real_Numbers/Proof_7 | [
"Exponential of Sum"
] | [
"Definition:Real Number",
"Definition:Exponential Function/Real"
] | [
"Definition:Exponential Function/Real/Limit of Sequence",
"Combination Theorem for Sequences/Real/Product Rule",
"Generalized Exponential Limit",
"Definition:Exponential Function/Real/Power Series Expansion",
"Equivalence of Definitions of Real Exponential Function/Limit of Sequence implies Power Series Exp... |
proofwiki-1685 | Exponential of Product | Let $x, y \in \R$ be real numbers.
Let $\exp x$ be the exponential of $x$.
Then:
:$\map \exp {x y} = \paren {\exp y}^x$ | Let $Y = \exp y$.
From Exponential of Natural Logarithm:
:$\map \ln {\exp y} = y$
From Logarithms of Powers, we have:
:$\ln Y^x = x \ln Y = x \, \map \ln {\exp y} = x y$
Thus:
:$\map \exp {x y} = \map \exp {\ln Y^x} = Y^x = \paren {\exp y}^x$
{{qed}} | Let $x, y \in \R$ be [[Definition:Real Number|real numbers]].
Let $\exp x$ be the [[Definition:Real Exponential Function|exponential of $x$]].
Then:
:$\map \exp {x y} = \paren {\exp y}^x$ | Let $Y = \exp y$.
From [[Exponential of Natural Logarithm]]:
:$\map \ln {\exp y} = y$
From [[Logarithms of Powers]], we have:
:$\ln Y^x = x \ln Y = x \, \map \ln {\exp y} = x y$
Thus:
:$\map \exp {x y} = \map \exp {\ln Y^x} = Y^x = \paren {\exp y}^x$
{{qed}} | Exponential of Product/Proof 1 | https://proofwiki.org/wiki/Exponential_of_Product | https://proofwiki.org/wiki/Exponential_of_Product/Proof_1 | [
"Exponential of Product",
"Exponential Function"
] | [
"Definition:Real Number",
"Definition:Exponential Function/Real"
] | [
"Exponential of Natural Logarithm",
"Logarithm of Power"
] |
proofwiki-1686 | Exponential of Product | Let $x, y \in \R$ be real numbers.
Let $\exp x$ be the exponential of $x$.
Then:
:$\map \exp {x y} = \paren {\exp y}^x$ | {{begin-eqn}}
{{eqn | l = \paren {\exp y}^x
| r = \map \exp {x \map \ln {\exp y} }
| c = {{Defof|Power to Real Number}}
}}
{{eqn | r = \map \exp {x y}
| c = Exponential of Natural Logarithm
}}
{{end-eqn}}
{{qed}} | Let $x, y \in \R$ be [[Definition:Real Number|real numbers]].
Let $\exp x$ be the [[Definition:Real Exponential Function|exponential of $x$]].
Then:
:$\map \exp {x y} = \paren {\exp y}^x$ | {{begin-eqn}}
{{eqn | l = \paren {\exp y}^x
| r = \map \exp {x \map \ln {\exp y} }
| c = {{Defof|Power to Real Number}}
}}
{{eqn | r = \map \exp {x y}
| c = [[Exponential of Natural Logarithm]]
}}
{{end-eqn}}
{{qed}} | Exponential of Product/Proof 2 | https://proofwiki.org/wiki/Exponential_of_Product | https://proofwiki.org/wiki/Exponential_of_Product/Proof_2 | [
"Exponential of Product",
"Exponential Function"
] | [
"Definition:Real Number",
"Definition:Exponential Function/Real"
] | [
"Exponential of Natural Logarithm"
] |
proofwiki-1687 | Exponential of Product | Let $x, y \in \R$ be real numbers.
Let $\exp x$ be the exponential of $x$.
Then:
:$\map \exp {x y} = \paren {\exp y}^x$ | For $n \in \Z_{\ge 0}$:
{{begin-eqn}}
{{eqn | l = \map \exp {n y}
| r = \map \exp {\sum_{k \mathop = 1}^n y}
}}
{{eqn | r = \prod_{k \mathop = 1}^n \exp y
| c = Exponential of Sum of Real Numbers
}}
{{eqn | n = 1
| r = \paren {\exp y}^n
}}
{{end-eqn}}
That is:
:$\forall n \in \Z_{\ge 0}: \map \exp {n ... | Let $x, y \in \R$ be [[Definition:Real Number|real numbers]].
Let $\exp x$ be the [[Definition:Real Exponential Function|exponential of $x$]].
Then:
:$\map \exp {x y} = \paren {\exp y}^x$ | For $n \in \Z_{\ge 0}$:
{{begin-eqn}}
{{eqn | l = \map \exp {n y}
| r = \map \exp {\sum_{k \mathop = 1}^n y}
}}
{{eqn | r = \prod_{k \mathop = 1}^n \exp y
| c = [[Exponential of Sum of Real Numbers]]
}}
{{eqn | n = 1
| r = \paren {\exp y}^n
}}
{{end-eqn}}
That is:
:$\forall n \in \Z_{\ge 0}: \map \... | Exponential of Product/Proof 3 | https://proofwiki.org/wiki/Exponential_of_Product | https://proofwiki.org/wiki/Exponential_of_Product/Proof_3 | [
"Exponential of Product",
"Exponential Function"
] | [
"Definition:Real Number",
"Definition:Exponential Function/Real"
] | [
"Exponential of Sum/Real Numbers",
"Reciprocal of Real Exponential",
"Real Number to Negative Power/Positive Integer",
"Definition:Integer",
"Definition:Strictly Positive/Integer",
"Definition:Power (Algebra)/Real Number/Definition 2"
] |
proofwiki-1688 | Exponent Combination Laws | Let $a, b \in \R_{>0}$ be strictly positive real numbers.
Let $x, y \in \R$ be real numbers.
Let $a^x$ be defined as $a$ to the power of $x$.
Then:
=== Product of Powers ===
{{:Exponent Combination Laws/Product of Powers}}
=== Power of Product ===
{{:Exponent Combination Laws/Power of Product}}
=== Negative 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 $a, b \in \R_{>0}$ be [[Definition:Strictly Positive Real Number|strictly positive real numbers]].
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:
=== [[Exponent Combination Laws/Product of Powers|Product o... | {{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 | https://proofwiki.org/wiki/Exponent_Combination_Laws/Power_of_Power/Proof_1 | [
"Exponent Combination Laws",
"Exponents",
"Powers",
"Analysis"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Real Number",
"Definition:Power (Algebra)/Real Number",
"Exponent Combination Laws/Product of Powers",
"Exponent Combination Laws/Power of Product",
"Exponent Combination Laws/Negative Power",
"Exponent Combination Laws/Power of Power",
"Exponent... | [
"Logarithm of Power"
] |
proofwiki-1689 | Exponent Combination Laws | Let $a, b \in \R_{>0}$ be strictly positive real numbers.
Let $x, y \in \R$ be real numbers.
Let $a^x$ be defined as $a$ to the power of $x$.
Then:
=== Product of Powers ===
{{:Exponent Combination Laws/Product of Powers}}
=== Power of Product ===
{{:Exponent Combination Laws/Power of Product}}
=== Negative 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 $a, b \in \R_{>0}$ be [[Definition:Strictly Positive Real Number|strictly positive real numbers]].
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:
=== [[Exponent Combination Laws/Product of Powers|Product o... | 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 | https://proofwiki.org/wiki/Exponent_Combination_Laws/Power_of_Power/Proof_2 | [
"Exponent Combination Laws",
"Exponents",
"Powers",
"Analysis"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Real Number",
"Definition:Power (Algebra)/Real Number",
"Exponent Combination Laws/Product of Powers",
"Exponent Combination Laws/Power of Product",
"Exponent Combination Laws/Negative Power",
"Exponent Combination Laws/Power of Power",
"Exponent... | [
"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-1690 | Exponent Combination Laws | Let $a, b \in \R_{>0}$ be strictly positive real numbers.
Let $x, y \in \R$ be real numbers.
Let $a^x$ be defined as $a$ to the power of $x$.
Then:
=== Product of Powers ===
{{:Exponent Combination Laws/Product of Powers}}
=== Power of Product ===
{{:Exponent Combination Laws/Power of Product}}
=== Negative Power ===
{... | {{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 $a, b \in \R_{>0}$ be [[Definition:Strictly Positive Real Number|strictly positive real numbers]].
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:
=== [[Exponent Combination Laws/Product of Powers|Product o... | {{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 | https://proofwiki.org/wiki/Exponent_Combination_Laws/Product_of_Powers/Proof_1 | [
"Exponent Combination Laws",
"Exponents",
"Powers",
"Analysis"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Real Number",
"Definition:Power (Algebra)/Real Number",
"Exponent Combination Laws/Product of Powers",
"Exponent Combination Laws/Power of Product",
"Exponent Combination Laws/Negative Power",
"Exponent Combination Laws/Power of Power",
"Exponent... | [
"Exponential of Sum"
] |
proofwiki-1691 | Exponent Combination Laws | Let $a, b \in \R_{>0}$ be strictly positive real numbers.
Let $x, y \in \R$ be real numbers.
Let $a^x$ be defined as $a$ to the power of $x$.
Then:
=== Product of Powers ===
{{:Exponent Combination Laws/Product of Powers}}
=== Power of Product ===
{{:Exponent Combination Laws/Power of Product}}
=== Negative Power ===
{... | 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 $a, b \in \R_{>0}$ be [[Definition:Strictly Positive Real Number|strictly positive real numbers]].
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:
=== [[Exponent Combination Laws/Product of Powers|Product o... | 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 | https://proofwiki.org/wiki/Exponent_Combination_Laws/Product_of_Powers/Proof_2 | [
"Exponent Combination Laws",
"Exponents",
"Powers",
"Analysis"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Real Number",
"Definition:Power (Algebra)/Real Number",
"Exponent Combination Laws/Product of Powers",
"Exponent Combination Laws/Power of Product",
"Exponent Combination Laws/Negative Power",
"Exponent Combination Laws/Power of Power",
"Exponent... | [
"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-1692 | Derivative of Exponential at Zero | Let $\exp x$ be the exponential of $x$ for real $x$.
Then:
:$\ds \lim_{x \mathop \to 0} \frac {\exp x - 1} x = 1$ | For all $x \in \R$:
:$\exp 0 - 1 = 0$ from Exponential of Zero
:$\map {D_x} {\exp x - 1} = \exp x$ from Sum Rule for Derivatives
:$D_x x = 1$ from Derivative of Identity Function.
Its prerequisites having been verified, {{Corollary|L'Hôpital's Rule|1}} yields immediately:
:$\ds \lim_{x \mathop \to 0} \frac {\exp x - 1}... | Let $\exp x$ be the [[Definition:Real Exponential Function|exponential]] of $x$ for [[Definition:Real Number|real]] $x$.
Then:
:$\ds \lim_{x \mathop \to 0} \frac {\exp x - 1} x = 1$ | For all $x \in \R$:
:$\exp 0 - 1 = 0$ from [[Exponential of Zero]]
:$\map {D_x} {\exp x - 1} = \exp x$ from [[Sum Rule for Derivatives]]
:$D_x x = 1$ from [[Derivative of Identity Function]].
Its prerequisites having been verified, {{Corollary|L'Hôpital's Rule|1}} yields immediately:
:$\ds \lim_{x \mathop \to 0} ... | Derivative of Exponential at Zero/Proof 1 | https://proofwiki.org/wiki/Derivative_of_Exponential_at_Zero | https://proofwiki.org/wiki/Derivative_of_Exponential_at_Zero/Proof_1 | [
"Derivatives involving Exponential Function",
"Derivative of Exponential at Zero"
] | [
"Definition:Exponential Function/Real",
"Definition:Real Number"
] | [
"Exponential of Zero",
"Sum Rule for Derivatives",
"Derivative of Identity Function"
] |
proofwiki-1693 | Derivative of Exponential at Zero | Let $\exp x$ be the exponential of $x$ for real $x$.
Then:
:$\ds \lim_{x \mathop \to 0} \frac {\exp x - 1} x = 1$ | Note that this proof does not presuppose Derivative of Exponential Function.
We use the definition of the exponential as a limit of a sequence:
{{begin-eqn}}
{{eqn | l = \frac {\exp h - 1} h
| r = \frac {\lim_{n \mathop \to \infty} \paren {1 + \dfrac h n}^n - 1} h
| c = {{Defof|Exponential Function/Real|sub... | Let $\exp x$ be the [[Definition:Real Exponential Function|exponential]] of $x$ for [[Definition:Real Number|real]] $x$.
Then:
:$\ds \lim_{x \mathop \to 0} \frac {\exp x - 1} x = 1$ | Note that this proof does not presuppose [[Derivative of Exponential Function]].
We use the definition of the exponential [[Definition:Exponential Function/Real/Limit of Sequence|as a limit of a sequence]]:
{{begin-eqn}}
{{eqn | l = \frac {\exp h - 1} h
| r = \frac {\lim_{n \mathop \to \infty} \paren {1 + \dfra... | Derivative of Exponential at Zero/Proof 2 | https://proofwiki.org/wiki/Derivative_of_Exponential_at_Zero | https://proofwiki.org/wiki/Derivative_of_Exponential_at_Zero/Proof_2 | [
"Derivatives involving Exponential Function",
"Derivative of Exponential at Zero"
] | [
"Definition:Exponential Function/Real",
"Definition:Real Number"
] | [
"Derivative of Exponential Function",
"Definition:Exponential Function/Real/Limit of Sequence",
"Binomial Theorem",
"Powers of Group Elements",
"Powers of Group Elements",
"Definition:Addition/Summand"
] |
proofwiki-1694 | Derivative of Exponential at Zero | Let $\exp x$ be the exponential of $x$ for real $x$.
Then:
:$\ds \lim_{x \mathop \to 0} \frac {\exp x - 1} x = 1$ | {{begin-eqn}}
{{eqn | l = \frac {e^x - 1} x
| r = \frac {e^x - e^0} x
| c = Exponential of Zero
}}
{{eqn | o = \to
| r = \valueat {\dfrac \d {\d x} e^x} {x \mathop = 0} {}
| c = {{Defof|Derivative of Real Function at Point}}, as $x \to 0$
}}
{{eqn | r = \valueat {e^x} {x \mathop = 0} {}
| ... | Let $\exp x$ be the [[Definition:Real Exponential Function|exponential]] of $x$ for [[Definition:Real Number|real]] $x$.
Then:
:$\ds \lim_{x \mathop \to 0} \frac {\exp x - 1} x = 1$ | {{begin-eqn}}
{{eqn | l = \frac {e^x - 1} x
| r = \frac {e^x - e^0} x
| c = [[Exponential of Zero]]
}}
{{eqn | o = \to
| r = \valueat {\dfrac \d {\d x} e^x} {x \mathop = 0} {}
| c = {{Defof|Derivative of Real Function at Point}}, as $x \to 0$
}}
{{eqn | r = \valueat {e^x} {x \mathop = 0} {}
... | Derivative of Exponential at Zero/Proof 3 | https://proofwiki.org/wiki/Derivative_of_Exponential_at_Zero | https://proofwiki.org/wiki/Derivative_of_Exponential_at_Zero/Proof_3 | [
"Derivatives involving Exponential Function",
"Derivative of Exponential at Zero"
] | [
"Definition:Exponential Function/Real",
"Definition:Real Number"
] | [
"Exponential of Zero",
"Derivative of Exponential Function",
"Exponential of Zero"
] |
proofwiki-1695 | Derivative of Exponential at Zero | Let $\exp x$ be the exponential of $x$ for real $x$.
Then:
:$\ds \lim_{x \mathop \to 0} \frac {\exp x - 1} x = 1$ | As we consider $x \to 0$, we may assume that $0 < \size x \le 1$.
Then:
{{begin-eqn}}
{{eqn | l = \size {\frac {e^x - 1} x - 1}
| r = \size {\frac {e^x - 1 - x} x}
}}
{{eqn | r = \size {\frac {\sum_{n \mathop = 0}^\infty \frac {x^n} {n!} - 1 - x} x }
| c = {{Defof|Exponential Function|subdef = Real/Power Se... | Let $\exp x$ be the [[Definition:Real Exponential Function|exponential]] of $x$ for [[Definition:Real Number|real]] $x$.
Then:
:$\ds \lim_{x \mathop \to 0} \frac {\exp x - 1} x = 1$ | As we consider $x \to 0$, we may assume that $0 < \size x \le 1$.
Then:
{{begin-eqn}}
{{eqn | l = \size {\frac {e^x - 1} x - 1}
| r = \size {\frac {e^x - 1 - x} x}
}}
{{eqn | r = \size {\frac {\sum_{n \mathop = 0}^\infty \frac {x^n} {n!} - 1 - x} x }
| c = {{Defof|Exponential Function|subdef = Real/Power S... | Derivative of Exponential at Zero/Proof 4 | https://proofwiki.org/wiki/Derivative_of_Exponential_at_Zero | https://proofwiki.org/wiki/Derivative_of_Exponential_at_Zero/Proof_4 | [
"Derivatives involving Exponential Function",
"Derivative of Exponential at Zero"
] | [
"Definition:Exponential Function/Real",
"Definition:Real Number"
] | [] |
proofwiki-1696 | Classification of Groups of Order up to 15 | Up to isomorphism, every group of order $\order G \le 15$ is one of the below:
{| class="sortable wikitable"
|- bgcolor="#ececec"
! Order !! Abelian !! Non-Abelian
|-
|1 || $\Z_1$ ||
|-
|2 || $\Z_2$ ||
|-
|3 || $\Z_3$ ||
|-
|4 || $\Z_4, \Z_2 \oplus \Z_2$ ||
|-
|5 || $\Z_5$ ||
|-
|6 || $\Z_6$ || $D_3 \cong S_3$
|-
|7... | The Abelian cases are the direct result of the Fundamental Theorem of Finite Abelian Groups.
The non-Abelian cases follow from seven separate theorems:
:$(1): \quad$ Trivial Group is Cyclic Group - determines theorem for order $1$
:$(2): \quad$ Prime Group is Cyclic - determines theorem for orders $2$, $3$, $5$, $7$, $... | Up to [[Definition:Group Isomorphism|isomorphism]], every [[Definition:Group|group]] of order $\order G \le 15$ is one of the below:
{| class="sortable wikitable"
|- bgcolor="#ececec"
! Order !! Abelian !! Non-Abelian
|-
|1 || $\Z_1$ ||
|-
|2 || $\Z_2$ ||
|-
|3 || $\Z_3$ ||
|-
|4 || $\Z_4, \Z_2 \oplus \Z_2$ ||
|-
|5... | The [[Definition:Abelian Group|Abelian]] cases are the direct result of the [[Fundamental Theorem of Finite Abelian Groups]].
The non-Abelian cases follow from seven separate theorems:
:$(1): \quad$ [[Trivial Group is Cyclic Group]] - determines theorem for order $1$
:$(2): \quad$ [[Prime Group is Cyclic]] - determin... | Classification of Groups of Order up to 15 | https://proofwiki.org/wiki/Classification_of_Groups_of_Order_up_to_15 | https://proofwiki.org/wiki/Classification_of_Groups_of_Order_up_to_15 | [
"Order of Groups"
] | [
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Definition:Group",
"Definition:Dihedral Group",
"Definition:Order of Structure",
"Definition:Symmetric Group",
"Definition:Alternating Group",
"Definition:Dicyclic Group",
"Definition:Order of Structure"
] | [
"Definition:Abelian Group",
"Fundamental Theorem of Finite Abelian Groups",
"Trivial Group is Cyclic Group",
"Prime Group is Cyclic",
"Group of Order Prime Squared is Abelian",
"Group of Order p q is Cyclic",
"Groups of Order 2p",
"Groups of Order 8",
"Groups of Order 12"
] |
proofwiki-1697 | Existence of Euler-Mascheroni Constant | The real sequence:
:$\ds \sequence {\sum_{k \mathop = 1}^n \frac 1 k - \ln n}$
converges to a limit.
This limit is known as the Euler-Mascheroni constant. | Let $f: \R \setminus \set 0 \to \R: \map f x = \dfrac 1 x$.
Clearly $f$ is continuous and positive on $\hointr 1 {+\infty}$.
From Reciprocal Sequence is Strictly Decreasing, $f$ is decreasing on $\hointr 1 {+\infty}$.
Therefore the conditions of the Cauchy Integral Test hold.
Thus the sequence $\sequence {\Delta_n}$ de... | The [[Definition:Real Sequence|real sequence]]:
:$\ds \sequence {\sum_{k \mathop = 1}^n \frac 1 k - \ln n}$
[[Definition:Convergent Real Sequence|converges]] to a [[Definition:Limit of Real Sequence|limit]].
This limit is known as the [[Definition:Euler-Mascheroni Constant|Euler-Mascheroni constant]]. | Let $f: \R \setminus \set 0 \to \R: \map f x = \dfrac 1 x$.
[[Definition:Clearly|Clearly]] $f$ is [[Definition:Continuous Real Function|continuous]] and [[Definition:Positive Real Function|positive]] on $\hointr 1 {+\infty}$.
From [[Reciprocal Sequence is Strictly Decreasing]], $f$ is [[Definition:Decreasing Real Fun... | Existence of Euler-Mascheroni Constant/Proof 1 | https://proofwiki.org/wiki/Existence_of_Euler-Mascheroni_Constant | https://proofwiki.org/wiki/Existence_of_Euler-Mascheroni_Constant/Proof_1 | [
"Existence of Euler-Mascheroni Constant",
"Euler-Mascheroni Constant"
] | [
"Definition:Real Sequence",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Limit of Sequence/Real Numbers",
"Definition:Euler-Mascheroni Constant"
] | [
"Definition:Clearly",
"Definition:Continuous Real Function",
"Definition:Positive Real Function",
"Reciprocal Sequence is Strictly Decreasing",
"Definition:Decreasing/Real Function",
"Cauchy Integral Test",
"Definition:Sequence",
"Definition:Decreasing/Sequence",
"Definition:Bounded Below Sequence",... |
proofwiki-1698 | Existence of Euler-Mascheroni Constant | The real sequence:
:$\ds \sequence {\sum_{k \mathop = 1}^n \frac 1 k - \ln n}$
converges to a limit.
This limit is known as the Euler-Mascheroni constant. | For $n \in \N_{>0}$ let:
:$\ds \gamma_n := \sum_{k \mathop = 1}^n \frac 1 k - \ln n$
Then:
{{begin-eqn}}
{{eqn | l = \gamma_n
| r = 1 + \int_1^n \dfrac {\floor u} {u^2} \rd u - \ln n
| c = Integral Expression of Harmonic Number
}}
{{eqn | r = 1 + \int_1^n \dfrac {\floor u} {u^2} \rd u - \int _1 ^n \dfrac ... | The [[Definition:Real Sequence|real sequence]]:
:$\ds \sequence {\sum_{k \mathop = 1}^n \frac 1 k - \ln n}$
[[Definition:Convergent Real Sequence|converges]] to a [[Definition:Limit of Real Sequence|limit]].
This limit is known as the [[Definition:Euler-Mascheroni Constant|Euler-Mascheroni constant]]. | For $n \in \N_{>0}$ let:
:$\ds \gamma_n := \sum_{k \mathop = 1}^n \frac 1 k - \ln n$
Then:
{{begin-eqn}}
{{eqn | l = \gamma_n
| r = 1 + \int_1^n \dfrac {\floor u} {u^2} \rd u - \ln n
| c = [[Integral Expression of Harmonic Number]]
}}
{{eqn | r = 1 + \int_1^n \dfrac {\floor u} {u^2} \rd u - \int _1 ^n \d... | Existence of Euler-Mascheroni Constant/Proof 2 | https://proofwiki.org/wiki/Existence_of_Euler-Mascheroni_Constant | https://proofwiki.org/wiki/Existence_of_Euler-Mascheroni_Constant/Proof_2 | [
"Existence of Euler-Mascheroni Constant",
"Euler-Mascheroni Constant"
] | [
"Definition:Real Sequence",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Limit of Sequence/Real Numbers",
"Definition:Euler-Mascheroni Constant"
] | [
"Integral Expression of Harmonic Number",
"Linear Combination of Integrals/Definite",
"Relative Sizes of Definite Integrals",
"Sum of Integrals on Adjacent Intervals for Integrable Functions",
"Monotone Convergence Theorem (Real Analysis)/Increasing Sequence",
"Definition:Real Sequence",
"Definition:Con... |
proofwiki-1699 | Group of Order p q is Cyclic | Let $p, q$ be primes such that $p < q$ and $p$ does not divide $q - 1$.
Let $G$ be a group of order $p q$.
Then $G$ is cyclic. | By Sylow $p$-Subgroup is Unique iff Normal, $H$ and $K$ are normal subgroups of $G$.
Let $H = \gen x$ and $K = \gen y$.
To show $G$ is cyclic, it is sufficient to show that $x$ and $y$ commute, because then:
:$\order {x y} = \order x \order y = p q$
where $\order x$ denotes the order of $x$ in $G$.
{{explain|Why does i... | Let $p, q$ be [[Definition:Prime Number|primes]] such that $p < q$ and $p$ does not [[Definition:Divisor of Integer|divide]] $q - 1$.
Let $G$ be a [[Definition:Group|group]] of [[Definition:Order of Structure|order]] $p q$.
Then $G$ is [[Definition:Cyclic Group|cyclic]]. | By [[Sylow p-Subgroup is Unique iff Normal|Sylow $p$-Subgroup is Unique iff Normal]], $H$ and $K$ are [[Definition:Normal Subgroup|normal subgroups]] of $G$.
Let $H = \gen x$ and $K = \gen y$.
To show $G$ is [[Definition:Cyclic Group|cyclic]], it is sufficient to show that $x$ and $y$ [[Definition:Commute|commute]], ... | Group of Order p q is Cyclic/Proof 1 | https://proofwiki.org/wiki/Group_of_Order_p_q_is_Cyclic | https://proofwiki.org/wiki/Group_of_Order_p_q_is_Cyclic/Proof_1 | [
"Group of Order p q is Cyclic",
"Finite Cyclic Groups",
"Groups of Order p q"
] | [
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer",
"Definition:Group",
"Definition:Order of Structure",
"Definition:Cyclic Group"
] | [
"Sylow p-Subgroup is Unique iff Normal",
"Definition:Normal Subgroup",
"Definition:Cyclic Group",
"Definition:Commutative/Elements",
"Definition:Order of Group Element",
"Definition:Normal Subgroup",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] |
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