TheoremBase

Theorems

A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.

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  • Cauchy Sequence of Real Numbers

    definitiondef:cauchy-sequence-real-c54-2026aAnalysis
    A sequence (an)n=1∞(a_n)_{n=1}^\infty of real numbers is called a Cauchy sequence if for every ε>0\varepsilon>0 there exists N∈NN\in\mathbb{N} such that for all integers m,n≥Nm,n\ge N, ∣an−am∣<ε.|a_n-a_m|<\varepsilon.

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    Authors ChatGPT-5.4, Aaron · Created

  • Limit of a Sequence of Real Numbers

    definitiondef:limit-sequence-real-c54-2026aAnalysis
    Let (an)n=1∞(a_n)_{n=1}^\infty be a sequence of real numbers, and let L∈RL\in\mathbb{R}. One says that (an)(a_n) converges to LL if for every ε>0\varepsilon>0 there exists N∈NN\in\mathbb{N} such that for all integers n≥Nn\ge N, ∣an−L∣<ε.|a_n-L|<\varepsilon. In that case one writes…

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    Authors ChatGPT-5.4, Aaron · Created

  • Continuity on a Closed Interval Implies Uniform Continuity

    lemmalem:heine-cantor-closed-interval-c54-2026aAnalysis
    Let a,b∈Ra,b\in\mathbb{R} with a<ba<b. If f:[a,b]→Rf:[a,b]\to\mathbb{R} is continuous on [a,b][a,b], then ff is uniformly continuous on [a,b][a,b].

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    Authors ChatGPT-5.4, Aaron · Created

  • Upper Sum and Lower Sum of a Function on a Partition

    definitiondef:upper-lower-sums-partition-c54-2026aAnalysis
    Let f:[a,b]→Rf:[a,b]\to\mathbb{R}, and let P:a=x0<x1<⋯<xn=bP:a=x_0<x_1<\cdots<x_n=b be a partition of [a,b][a,b]. For each i=1,…,ni=1,\dots,n, let Mi=sup⁡{f(t):t∈[xi−1,xi]},mi=\reftextdef:lower−bound−infimum−c54−2026ainf⁡{f(t):t∈[xi−1,xi]}.M_i=\sup\{f(t): t\in[x_{i-1},x_i]\},\qquad m_i=\reftext{def:lower-bound-infimum-c54-2026a}{\inf}\{f(t): t\in[x_{i-1},x_i]\}. The upper sum and lower sum…

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    Authors ChatGPT-5.4, Aaron · Created

  • Least Upper Bound Property of the Real Numbers

    definitiondef:least-upper-bound-property-c54-2026aAnalysis
    The real numbers have the least upper bound property: whenever S⊆RS\subseteq\mathbb{R} is nonempty and bounded above, there exists a number u∈Ru\in\mathbb{R} such that u=sup⁡Su=\sup S.

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    Authors ChatGPT-5.4, Aaron · Created

  • Lower Bound and Greatest Lower Bound

    definitiondef:lower-bound-infimum-c54-2026aAnalysis
    Let S⊆RS\subseteq\mathbb{R}. A number ℓ∈R\ell\in\mathbb{R} is called a lower bound of SS if ℓ≤s\ell\le s for every s∈Ss\in S. If SS is nonempty and bounded below, then a number m∈Rm\in\mathbb{R} is called the greatest lower bound, or infimum, of SS if:…

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    Authors ChatGPT-5.4, Aaron · Created

  • Uniform Continuity on a Subset of the Real Numbers

    definitiondef:uniform-continuity-real-subset-c54-2026aAnalysis
    Let E⊆RE\subseteq\mathbb{R} and let f:E→Rf:E\to\mathbb{R}. The function ff is said to be uniformly continuous on EE if for every ε>0\varepsilon>0 there exists δ>0\delta>0 such that for all x,y∈Ex,y\in E, if ∣x−y∣<δ|x-y|<\delta, then ∣f(x)−f(y)∣<ε.|f(x)-f(y)|<\varepsilon.

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    Authors ChatGPT-5.4, Aaron · Created

  • Additivity of the Riemann Integral on Adjacent Intervals

    lemmalem:riemann-integral-additivity-adjacent-intervals-c54-2026aAnalysis
    Let a,b,c∈Ra,b,c\in\mathbb{R} satisfy a≤b≤ca\le b\le c, and let f:[a,c]→Rf:[a,c]\to\mathbb{R} be Riemann integrable on [a,c][a,c]. Then the restrictions f∣[a,b]:[a,b]→Rf|_{[a,b]}:[a,b]\to\mathbb{R} and f∣[b,c]:[b,c]→Rf|_{[b,c]}:[b,c]\to\mathbb{R} are Riemann integrable on [a,b][a,b] and [b,c][b,c], respectively, and…

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    Authors ChatGPT-5.4, Aaron · Created

  • Riemann Sum on a Tagged Partition

    definitiondef:riemann-sum-tagged-partition-c54-2026aAnalysis
    Let a,b∈Ra,b\in\mathbb{R} with a<ba<b, let f:[a,b]→Rf:[a,b]\to\mathbb{R}, let P=(x0,x1,…,xn)P=(x_0,x_1,\dots,x_n) be a partition of [a,b][a,b], and let (t1,…,tn)(t_1,\dots,t_n) determine a tagged partition of [a,b][a,b] relative to PP. The corresponding Riemann sum of ff is the real number…

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    Authors ChatGPT-5.4, Aaron · Created

  • Tagged Partition of a Closed Interval

    definitiondef:tagged-partition-closed-interval-c54-2026aAnalysis
    Let a,b∈Ra,b\in\mathbb{R} with a<ba<b, and let P=(x0,x1,…,xn)P=(x_0,x_1,\dots,x_n) be a partition of [a,b][a,b]. A tagged partition of [a,b][a,b] relative to PP is a choice of points ti∈[xi−1,xi]t_i\in[x_{i-1},x_i] for each i=1,…,ni=1,\dots,n.

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    Authors ChatGPT-5.4, Aaron · Created

  • 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 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. 1. For every x∈Sx\in S, one has x≤xx\le x. [Reflexivi…

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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. 1. 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] 2. There exists a…

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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 Ordered Field 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 Upper Bound and Least Upper Bound.

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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 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…

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    Authors ChatGPT-5.4, Aaron · Created

  • Ordered Field

    definitiondef:ordered-field-c54-2026bAnalysisAlgebra
    An ordered field is a field FF together with a binary relation ≤\le on FF such that ≤\le is a total order on FF, and the order is compatible with the field operations in the following sense. 1. For all a,b,c∈Fa,b,c\in F, if a≤ba\le b, then a+c≤b+ca+c\le b+c. 2. For all a,b∈Fa,b\in F, if…

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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 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…

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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 Interval in the Real Line, 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 Continuity on a Closed Interval, and let F:I→RF:I\to\mathbb{R} be an antiderivative of ff on II in the sense of…

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    Authors ChatGPT-5.4, Aaron · Created

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