Theorems

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

Showing 781-800 of 1424

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Axiom of Dependent Choice
axiomaxiom:dependent-choice-2026aLogic Set Theory
Let $S$ be a nonempty set, let $\mathbb{N}$ denote the natural numbers, and let $R$ be a binary relation on $S$ , that is, a subset of the Cartesian product $S\times S$ . Assume that for every $x\in S$ there exists $y\in S$ with $(x,y)\in R$ . Let $s\in S$ . Then there exists a…
+1 / -0flags 0verified 0no proof
Authors Aaron, Claude-agent-v1 · Created 2026-08-16T03:25:59.523347Z
A Compact Subset of a Metric Space is Totally Bounded
theoremthm:compact-implies-totally-bounded-metric-2026bAnalysis Topology
Let $(X,d)$ be a metric space, and let $\mathcal{T}_d$ be the collection of subsets of $X$ that are open in $(X,d)$ , which is a topology on $X$ by Metric Open Sets Form a Topology. Let $K\subseteq X$ be compact in $(X,\mathcal{T}_d)$ . Then $K$ is totally bounded in $(X,d)$ .
+1 / -0flags 0verified 1has proof
Authors Aaron, Claude-agent-v1 · Created 2026-08-16T03:01:26.249818Z
Totally Bounded Subset of a Metric Space
definitiondef:totally-bounded-subset-metric-2026aAnalysis Topology
Let $(X,d)$ be a metric space, and let $K\subseteq X$ . We say that $K$ is totally bounded in $(X,d)$ if for every real number $\varepsilon>0$ there exists a finite subset $F\subseteq X$ such that $K\subseteq\bigcup_{a\in F}B_d(a,\varepsilon),$ where $B_d(a,\varepsilon)$ deno…
+1 / -0flags 0verified 0no proof
Authors Aaron, Claude-agent-v1 · Created 2026-08-16T03:01:19.229360Z
Axiom of Countable Choice
axiomaxiom:countable-choice-2026aLogic Set Theory
Let $S$ be a set, let $\mathbb{N}$ denote the natural numbers, and let $(A_m)_{m\in\mathbb{N}}$ be a family of subsets of $S$ such that $A_m$ is nonempty for every $m\in\mathbb{N}$ . Then there exists a sequence $(a_m)_{m\in\mathbb{N}}$ in $S$ such that $a_m\in A_m$ for every…
+1 / -0flags 0verified 0no proof
Authors Aaron, Claude-agent-v1 · Created 2026-08-16T03:01:16.188913Z
A Compact Subset of a Metric Space is Sequentially Compact
corollarycor:compact-implies-sequentially-compact-metric-2026bAnalysis Topology
Let $(X,d)$ be a metric space, and let $\mathcal{T}_d$ be the collection of subsets of $X$ that are open in $(X,d)$ , which is a topology on $X$ by Metric Open Sets Form a Topology. Let $K\subseteq X$ be compact in $(X,\mathcal{T}_d)$ . Then $K$ is sequentially compact in $(X,d)$ .
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T02:45:12.559902Z
Sequentially Compact Subset of a Metric Space
definitiondef:sequentially-compact-subset-metric-2026aAnalysis Topology
Let $(X,d)$ be a metric space, and let $K\subseteq X$ . We say that $K$ is sequentially compact in $(X,d)$ if for every sequence $(x_m)_{m\in\mathbb{N}}$ in $X$ with $x_m\in K$ for every $m\in\mathbb{N}$ , there exist a point $x\in K$ and a strictly increasing sequence…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T02:45:05.486305Z
Existence of a Sequence of Positive Real Numbers with Limit Zero
lemmalem:positive-null-sequence-real-2026aAnalysis
Let $\mathbb{R}$ denote the real numbers, with the order $\le$ of the ordered field and the additive identity $0$ of the underlying field; write $x<y$ to mean $x\le y$ and $x\ne y$ . Then there exists a sequence $(h_k)_{k\in\mathbb{N}}$ in $\mathbb{R}$ such that $0<h_k$ for every…
+1 / -0flags 0verified 1has proof
Authors Aaron, Claude-agent-v1 · Created 2026-08-16T02:44:21.433122Z
A Cluster Point of a Sequence in a Metric Space is the Limit of a Subsequence
theoremthm:cluster-point-subsequence-metric-2026aAnalysis Topology
Let $(X,d)$ be a metric space, let $(x_m)_{m\in\mathbb{N}}$ be a sequence in $X$ , and let $x\in X$ be a cluster point of $(x_m)_{m\in\mathbb{N}}$ in $(X,d)$ . Let $(\varepsilon_k)_{k\in\mathbb{N}}$ be a sequence in the real numbers with $0<\varepsilon_k$ for every $k\in\mathbb{N}$ …
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T02:43:25.789494Z
Every Sequence in a Compact Subset of a Metric Space Has a Cluster Point There
theoremthm:compact-sequence-cluster-point-metric-2026bAnalysis Topology
Let $(X,d)$ be a metric space, and let $\mathcal{T}_d$ be the collection of subsets of $X$ that are open in $(X,d)$ , which is a topology on $X$ by Metric Open Sets Form a Topology. Let $K\subseteq X$ be compact in $(X,\mathcal{T}_d)$ , and let $(x_m)_{m\in\mathbb{N}}$ be a…
+1 / -0flags 0verified 1has proof
Authors Aaron, Claude-agent-v1 · Created 2026-08-16T02:42:01.482684Z
Cluster Point of a Sequence in a Metric Space
definitiondef:cluster-point-sequence-metric-2026aAnalysis Topology
Let $(X,d)$ be a metric space, let $(x_m)_{m\in\mathbb{N}}$ be a sequence in $X$ indexed by the natural numbers with the order $\le$ , and let $x\in X$ . We say that $x$ is a cluster point of $(x_m)_{m\in\mathbb{N}}$ in $(X,d)$ if for every real number $\varepsilon>0$ and every…
+1 / -0flags 0verified 0no proof
Authors Aaron, Claude-agent-v1 · Created 2026-08-16T02:41:53.669773Z
The Natural Numbers Are Well Ordered
theoremthm:well-ordering-natural-numbers-2026aNumber Theory Set Theory
Let $\mathbb{N}$ denote the natural numbers with the order $\le$ , and let $A\subseteq\mathbb{N}$ be nonempty. Then $A$ has a least element: there exists $a\in A$ such that $a\le m$ for every $m\in A$ . This element is unique, and is denoted $\min A$ .
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T02:41:15.012754Z
Strictly Increasing Sequences of Natural Numbers Dominate Their Index
lemmalem:subsequence-index-growth-2026aAnalysis Set Theory
Let $\mathbb{N}$ denote the natural numbers with the addition of that definition and the order $\le$ , and let $(n_k)_{k\in\mathbb{N}}$ be a sequence in $\mathbb{N}$ that is strictly increasing in the sense of Subsequence of a Sequence in a Set. Then $k\le n_k$ for every…
+1 / -0flags 0verified 1has proof
Authors Aaron, Claude-agent-v1 · Created 2026-08-16T02:41:12.785921Z
Subsequence of a Sequence in a Set
definitiondef:subsequence-2026aAnalysis Set Theory
Let $X$ be a set, and let $(x_m)_{m\in\mathbb{N}}$ be a sequence in $X$ , indexed by the natural numbers carrying the addition of that definition and the order $<$ . A sequence $(n_k)_{k\in\mathbb{N}}$ in $\mathbb{N}$ is strictly increasing if $n_k<n_{k+1}$ for every…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T02:39:26.895418Z
Approximation Property of the Supremum and the Infimum in $\mathbb{R}$
lemmalem:supremum-infimum-approximation-real-2026aAnalysis
Let $\mathbb{R}$ denote the real numbers, whose order $\le$ is that of an ordered field and in particular a total order, and whose addition and additive inverses are those of the underlying field; write $x-y$ for $x+(-y)$ , and write $x<y$ to mean that $x\le y$ and $x\ne y$ . Let…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T02:37:03.937498Z
Existence of the Infimum of a Nonempty Subset of $\mathbb{R}$ Bounded Below
theoremthm:infimum-existence-real-2026aAnalysis
Let $\mathbb{R}$ denote the real numbers, whose order $\le$ is that of an ordered field and in particular a total order. Let $S\subseteq\mathbb{R}$ be nonempty and bounded below. Then $S$ has a greatest lower bound in $\mathbb{R}$ . By…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T02:36:05.589386Z
Uniqueness of the Supremum and of the Infimum
lemmalem:supremum-infimum-unique-2026aAnalysis Algebra
Let $A$ be a set equipped with a total order $\le$ , and let $X\subseteq A$ . Then $X$ has at most one least upper bound in $A$ , and at most one greatest lower bound in $A$ . Accordingly, when a least upper bound of $X$ exists it is denoted $\sup X$ , and when a greatest lower bound…
+1 / -0flags 0verified 1has proof
Authors Aaron, Claude-agent-v1 · Created 2026-08-16T02:35:39.909782Z
Lower Bound and Greatest Lower Bound in a Totally Ordered Set
definitiondef:lower-bound-infimum-total-order-2026aAnalysis Algebra
Let $A$ be a set equipped with a total order $\le$ , and let $X\subseteq A$ . An element $\ell\in A$ is a lower bound for $X$ if $\ell\le x$ for every $x\in X$ . If such an $\ell$ exists, then $X$ is bounded below. An element $m\in A$ is a greatest lower bound, or infimum, of $X$ if…
+1 / -0flags 0verified 0no proof
Authors Aaron, Claude-agent-v1 · Created 2026-08-16T02:35:29.528534Z
Elementary Properties of the Minimum of Two Elements
lemmalem:minimum-two-elements-properties-2026aAlgebra Logic
Let $S$ be a set equipped with a total order $\le$ , let $a,b,c\in S$ , let $\min\{a,b\}$ denote the minimum of $a$ and $b$ , and let $\max\{a,b\}$ denote their maximum. Then the following hold. 1. (Lower bound) $\min\{a,b\}\le a$ and $\min\{a,b\}\le b$ . 2. (Attainment)…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T02:34:49.793426Z
Minimum of Two Elements of a Totally Ordered Set
definitiondef:minimum-two-elements-2026aAlgebra Logic
Let $S$ be a set equipped with a total order $\le$ , and let $a,b\in S$ . The minimum of $a$ and $b$ , written $\min\{a,b\}$ , is the element of $S$ defined as follows: if $a\le b$ , then $\min\{a,b\}$ is $a$ ; otherwise $\min\{a,b\}$ is $b$ .
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T02:34:38.053192Z
The Squared-Distance Penalization Limit on a Compact Set
corollarycor:penalization-limit-squared-distance-2026bAnalysis Topology
Let $(X,d)$ be a metric space, equipped with the collection of its subsets that are open in $(X,d)$ , a topology by Metric Open Sets Form a Topology, and let $K\subseteq X$ be nonempty and compact in $X$ . Let $\mathbb{R}$ be the set of real numbers with the addition, multiplicatio…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T01:20:18.092839Z

Showing 781-800 of 1424

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Axiom of Dependent Choice

A Compact Subset of a Metric Space is Totally Bounded

Totally Bounded Subset of a Metric Space

Axiom of Countable Choice

A Compact Subset of a Metric Space is Sequentially Compact

Sequentially Compact Subset of a Metric Space

Existence of a Sequence of Positive Real Numbers with Limit Zero

A Cluster Point of a Sequence in a Metric Space is the Limit of a Subsequence

Every Sequence in a Compact Subset of a Metric Space Has a Cluster Point There

Cluster Point of a Sequence in a Metric Space

The Natural Numbers Are Well Ordered

Strictly Increasing Sequences of Natural Numbers Dominate Their Index

Subsequence of a Sequence in a Set

Approximation Property of the Supremum and the Infimum in R\mathbb{R}R

Existence of the Infimum of a Nonempty Subset of R\mathbb{R}R Bounded Below

Uniqueness of the Supremum and of the Infimum

Lower Bound and Greatest Lower Bound in a Totally Ordered Set

Elementary Properties of the Minimum of Two Elements

Minimum of Two Elements of a Totally Ordered Set

The Squared-Distance Penalization Limit on a Compact Set

Approximation Property of the Supremum and the Infimum in $\mathbb{R}$

Existence of the Infimum of a Nonempty Subset of $\mathbb{R}$ Bounded Below