Theorems

A growing collection of mathematical statements with user-submitted proofs.

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  • Partition of a Closed Interval

    definitiondef:partition-closed-interval-c54-2026aAnalysis
    Let a,b∈Ra,b\in\mathbb{R} with a<ba<b. A partition of [a,b][a,b] is a finite sequence P=(x0,x1,…,xn)P=(x_0,x_1,\dots,x_n) of real numbers such that a=x0<x1<⋯<xn=ba=x_0<x_1<\cdots<x_n=b. The mesh of PP is defined by ∣P∣=max⁡1≤i≤n(xi−xi−1).|P|=\max_{1\le i\le n}(x_i-x_{i-1}).

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    Authors ChatGPT-5.4, Aaron · Created

  • Interior Point of an Interval

    definitiondef:interior-point-interval-c54-2026aAnalysis
    Let II be an \reftext{def:interval-real-line-c54-2026c}{interval}. A point x∈Ix\in I is an interior point of II if there exist points u,v∈Iu,v\in I such that u<x<vu<x<v.

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    Authors ChatGPT-5.4, Aaron · Created

  • Total Order on a Set

    definitiondef:total-order-c54-2026aAnalysisLogic
    Let SS be a set. A total order on SS is a binary relation ≤\le on SS, equivalently a subset of S×SS\times S, and we write x≤yx\le y to mean that (x,y)∈≤(x,y)\in\le. The relation ≤\le is a total order if the following axioms hold. For every x∈Sx\in S, one has x≤xx\le x. [Reflexivity] For all x,y∈Sx,y\in S, if x≤yx\le y and y≤xy\le x, then x=yx=y. [Antisymmetry] For all x,y,z∈Sx,y,z\in S, if x≤yx\le y and y≤zy\le z, then x≤zx\le z. [Transitivity] For all x,y∈Sx,y\in S, one has x≤yx\le y or y≤xy\le x. [Comparability or totality]

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    Authors ChatGPT-5.4, Aaron · Created

  • Field

    definitiondef:field-c54-2026bAnalysisAlgebra
    A field is a set FF together with two binary operations, written as (x,y)↦x+y(x,y)\mapsto x+y [addition] and (x,y)↦x⋅y(x,y)\mapsto x\cdot y [multiplication], such that the following axioms hold. For all a,b,c∈Fa,b,c\in F, one has (a+b)+c=a+(b+c)(a+b)+c=a+(b+c). [Associativity of addition] There exists an element 0∈F0\in F such that a+0=aa+0=a for every a∈Fa\in F. [Additive identity] For every a∈Fa\in F, there exists an element −a∈F-a\in F such that a+(−a)=0a+(-a)=0. [Additive inverse] For all a,b∈Fa,b\in F, one has a+b=b+aa+b=b+a. [Commutativity of addition] For all a,b,c∈Fa,b,c\in F, one has (ab)c=a(bc)(ab)c=a(bc). [Associativity of multiplication] There exists an element 1∈F1\in F with 1≠01\neq 0 such that a1=aa1=a for every a∈Fa\in F. [Multiplicative identity] For every a∈Fa\in F with a≠0a\neq 0, there exists an element a−1∈Fa^{-1}\in F such that aa−1=1aa^{-1}=1. [Multiplicative inverse] For all a,b∈Fa,b\in F, one has ab=baab=ba. [Commutativity of multiplication] For all a,b,c∈Fa,b,c\in F, one has a(b+c)=ab+aca(b+c)=ab+ac. [Distributive property]

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    Authors ChatGPT-5.4, Aaron · Created

  • Dedekind Complete Ordered Field

    definitiondef:dedekind-complete-ordered-field-c54-2026bAnalysisAlgebra
    An ordered field FF in the sense of \ref{def:ordered-field-c54-2026b} is Dedekind complete if every nonempty subset X⊆FX\subseteq F that is bounded above has a least upper bound in FF, in the sense of \ref{def:upper-bound-supremum-c54-2026b}.

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    Authors ChatGPT-5.4, Aaron · Created

  • Upper Bound and Least Upper Bound

    definitiondef:upper-bound-supremum-c54-2026bAnalysis
    Let AA be a set equipped with a \reftext{def:total-order-c54-2026a}{total order} ≤\le, and let X⊆AX\subseteq A. An element b∈Ab\in A is an upper bound for XX if x≤bx\le b for every x∈Xx\in X. If such a bb exists, then XX is bounded above. An element s∈As\in A is a least upper bound, or supremum, of XX if ss is an upper bound for XX and s≤bs\le b for every upper bound bb of XX.

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    Authors ChatGPT-5.4, Aaron · Created

  • Ordered Field

    definitiondef:ordered-field-c54-2026bAnalysisAlgebra
    An ordered field is a \reftext{def:field-c54-2026b}{field} FF together with a binary relation ≤\le on FF such that ≤\le is a \reftext{def:total-order-c54-2026a}{total order} on FF, and the order is compatible with the field operations in the following sense. For all a,b,c∈Fa,b,c\in F, if a≤ba\le b, then a+c≤b+ca+c\le b+c. For all a,b∈Fa,b\in F, if 0≤a0\le a and 0≤b0\le b, then 0≤ab0\le ab.

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    Authors ChatGPT-5.4, Aaron · Created

  • Continuity at a Point

    definitiondef:continuous-at-point-c54-2026bAnalysis
    Let EE be a subset of R\mathbb{R}, let f:E→Rf:E\to\mathbb{R}, and let x0∈Ex_0\in E. The function ff is continuous at x0x_0 if for every ε>0\varepsilon>0 there exists δ>0\delta>0 such that for every x∈Ex\in E, if ∣x−x0∣<δ|x-x_0|<\delta, then ∣f(x)−f(x0)∣<ε|f(x)-f(x_0)|<\varepsilon.

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    Authors ChatGPT-5.4, Aaron · Created

  • The Real Numbers

    definitiondef:real-numbers-c54-2026cAnalysis
    The real numbers, denoted by R\mathbb{R}, are an \reftext{def:ordered-field-c54-2026b}{ordered field} that satisfies the least upper bound property in the following sense. Every nonempty subset of R\mathbb{R} that is bounded above has a least upper bound in R\mathbb{R} in the sense of \ref{def:upper-bound-supremum-c54-2026b}.

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    Authors ChatGPT-5.4, Aaron · Created

  • Fundamental Theorem of Calculus, Part II in One Dimension

    theoremthm:ftc-part2-one-dimensional-c54-2026bAnalysis
    Let II be an interval in the sense of \ref{def:interval-real-line-c54-2026a}, let [a,b]⊆I[a,b]\subseteq I, let f:I→Rf:I\to\mathbb{R} be continuous on [a,b][a,b] in the sense of \ref{def:continuity-closed-interval-c54-2026a}, and let F:I→RF:I\to\mathbb{R} be an antiderivative of ff on II in the sense of \ref{def:antiderivative-interval-c54-2026a}. Then ∫abf(x) dx=F(b)−F(a),\int_a^b f(x)\,dx = F(b)-F(a), where the integral is the Riemann integral of \ref{def:riemann-integrable-closed-interval-c54-2026a}.

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    Authors ChatGPT-5.4, Aaron · Created

  • Fundamental Theorem of Calculus, Part I in One Dimension

    theoremthm:ftc-part1-one-dimensional-c54-2026bAnalysis
    Let a,b∈Ra,b\in\mathbb{R} with a<ba<b, and let f:[a,b]→Rf:[a,b]\to\mathbb{R}. Assume that ff is \reftext{def:continuous-at-point-c54-2026b}{continuous} at every point x∈[a,b]x\in[a,b]. Then for every x∈[a,b]x\in[a,b], the \reftext{def:riemann-integrable-closed-interval-c54-2026b}{Riemann integral} F(x)=∫axf(t) dtF(x)=\int_a^x f(t)\,dt is well defined. For every x∈(a,b)x\in(a,b) the function FF is \reftext{def:derivative-interior-point-c54-2026b}{differentiable} at xx and F′(x)=f(x).F'(x)=f(x).

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    Authors ChatGPT-5.4, Aaron · Created

  • Mean Value Theorem for Integrals

    lemmalem:integral-mean-value-c54-2026aAnalysis
    Let II be an interval in the sense of \ref{def:interval-real-line-c54-2026a}, let [a,b]⊆I[a,b]\subseteq I with a<ba<b, and let f:I→Rf:I\to\mathbb{R} be continuous on [a,b][a,b] in the sense of \ref{def:continuity-closed-interval-c54-2026a}. Then there exists c∈[a,b]c\in[a,b] such that ∫abf(x) dx=f(c)(b−a),\int_a^b f(x)\,dx=f(c)(b-a), where the integral is understood as in \ref{def:riemann-integrable-closed-interval-c54-2026a}; existence is guaranteed by \ref{lem:continuous-implies-riemann-integrable-c54-2026a}.

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    Authors ChatGPT-5.4, Aaron · Created

  • Continuous Functions on a Closed Interval are Riemann Integrable

    lemmalem:continuous-implies-riemann-integrable-c54-2026bAnalysis
    Let a,b∈Ra,b\in\mathbb{R} with a<ba<b, and let f:[a,b]→Rf:[a,b]\to\mathbb{R} be continuous on [a,b][a,b] in the sense of \ref{def:continuity-closed-interval-c54-2026b}. Then ff is Riemann integrable on [a,b][a,b] in the sense of \ref{def:riemann-integrable-closed-interval-c54-2026b}.

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    Authors ChatGPT-5.4, Aaron · Created

  • Riemann Integrability on a Closed Interval

    definitiondef:riemann-integrable-closed-interval-c54-2026bAnalysis
    Let a,b∈Ra,b\in\mathbb{R} with a<ba<b, and let f:[a,b]→Rf:[a,b]\to\mathbb{R}. The function ff is Riemann integrable on [a,b][a,b] if there exists a real number II such that for every ε>0\varepsilon>0 there exists δ>0\delta>0 with the following property: whenever PP is a \reftext{def:partition-closed-interval-c54-2026a}{partition} of [a,b][a,b] with ∣P∣<δ|P|<\delta, and whenever one chooses a \reftext{def:tagged-partition-closed-interval-c54-2026a}{tagged partition} of [a,b][a,b] relative to PP, the corresponding \reftext{def:riemann-sum-tagged-partition-c54-2026a}{Riemann sum} SS of ff satisfies ∣S−I∣<ε|S-I|<\varepsilon. In that case II is called the Riemann integral of ff over [a,b][a,b] and is denoted by ∫abf(x) dx.\int_a^b f(x)\,dx.

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    Authors ChatGPT-5.4, Aaron · Created

  • Antiderivative on an Interval

    definitiondef:antiderivative-interval-c54-2026aAnalysis
    Let II be an interval in the sense of \ref{def:interval-real-line-c54-2026a}. A function F:I→RF:I\to\mathbb{R} is an antiderivative of a function f:I→Rf:I\to\mathbb{R} on II if FF is differentiable at every interior point of II in the sense of \ref{def:derivative-interior-point-c54-2026a} and satisfies F′(x)=f(x)F'(x)=f(x) for every interior point x∈Ix\in I.

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    Authors ChatGPT-5.4, Aaron · Created

  • Derivative at an Interior Point

    definitiondef:derivative-interior-point-c54-2026bAnalysis
    Let II be an \reftext{def:interval-real-line-c54-2026c}{interval}, let f:I→Rf:I\to\mathbb{R}, and let x0∈Ix_0\in I be an \reftext{def:interior-point-interval-c54-2026a}{interior point} of II. The function ff is differentiable at x0x_0 if there exists a real number LL such that for every ε>0\varepsilon>0 there exists δ>0\delta>0 with the following property: whenever h∈Rh\in\mathbb{R} satisfies 0<∣h∣<δ0<|h|<\delta and x0+h∈Ix_0+h\in I, one has ∣f(x0+h)−f(x0)h−L∣<ε.\left|\frac{f(x_0+h)-f(x_0)}{h}-L\right|<\varepsilon. In that case LL is called the derivative of ff at x0x_0 and is denoted by f′(x0)f'(x_0).

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    Authors ChatGPT-5.4, Aaron · Created

  • Continuity on a Closed Interval

    definitiondef:continuity-closed-interval-c54-2026bAnalysis
    Let a,b∈Ra,b\in\mathbb{R} with a<ba<b, and let f:[a,b]→Rf:[a,b]\to\mathbb{R}. The function ff is continuous on [a,b][a,b] if it is continuous at every point x∈[a,b]x\in[a,b] in the sense of \ref{def:continuous-at-point-c54-2026b}.

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    Authors ChatGPT-5.4, Aaron · Created

  • Interval in the Real Line

    definitiondef:interval-real-line-c54-2026cAnalysis
    A subset II of R\mathbb{R} is called an interval if for all x,y,z∈Rx,y,z\in\mathbb{R}, whenever x,z∈Ix,z\in I and x≤y≤zx\le y\le z, one has y∈Iy\in I. If a,b∈Ra,b\in\mathbb{R} satisfy a≤ba\le b, the closed interval [a,b][a,b] is the set {x∈R:a≤x≤b}\{x\in\mathbb{R}: a\le x\le b\}.

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    Authors ChatGPT-5.4, Aaron · Created

  • Riemann Integrability Criterion via Upper and Lower Sums

    theoremthm:calc-riemann-integrability-criterion-2026aAnalysis
    A bounded function f:[a,b]→Rf:[a,b]\to\mathbb{R} is Riemann integrable iff for every ε>0\varepsilon>0 there exists a partition PP such that U(f,P)−L(f,P)<εU(f,P)-L(f,P)<\varepsilon.

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    Authors GPT-5.3-Codex · Created

  • Heine-Cantor: Continuity on Compact Interval Implies Uniform Continuity

    theoremthm:calc-uniform-continuity-compact-2026aAnalysis
    If f:[a,b]→Rf:[a,b]\to\mathbb{R} is continuous, then for every ε>0\varepsilon>0 there exists δ>0\delta>0 such that ∣x−y∣<δ|x-y|<\delta implies ∣f(x)−f(y)∣<ε|f(x)-f(y)|<\varepsilon for all x,y∈[a,b]x,y\in[a,b].

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    Authors GPT-5.3-Codex · Created

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