LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passverified 2026-08-09 (gpt-5.6-terra-codex-subscription)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A Cauchy filter with a cluster point converges to that point

Statement

A Cauchy filter on a uniform space with a cluster point $x$ converges to $x$ .

Facts & Assumptions

Given: A Cauchy filter $F$ and one of its cluster points $x$ .

[L1]

A cluster point meets every filter member in every neighbourhood, while convergence means containment of every neighbourhood (Convergence and cluster points of a filter on a topological space).

[L2]

Cauchy filters have small members, and symmetric entourages have symmetric square roots (Cauchy filter in a uniform space, Every uniformity has a base of symmetric entourages).

[L3]

For the topology induced by a uniformity, every entourage ball $D [x]$ is a neighbourhood of $x$ (The sets containing an entourage ball about each of their points form a topology).

Proof

technique · direct

1.1

Let $E$ be an entourage and choose symmetric $D$ with $D \circ D \subseteq E$ ; choose $A \in F$ with $A \times A \subseteq D$ .

L2choose

1.2

The ball $D [x]$ is a neighbourhood of $x$ , so it meets $A$ because $x$ is a cluster point; fix $a \in A \cap D [x]$ .

L1L3choose

2.1

For every $b \in A$ , symmetry gives $(x, a) \in D$ and smallness gives $(a, b) \in D$ , hence $(x, b) \in E$ ; so $A \subseteq E [x]$ .

step 1.1step 1.2

3.1

Every entourage ball about $x$ belongs to $F$ by upward closure, and such balls form a neighbourhood base at $x$ , so $F$ converges to $x$ .

step 2.1L1L3∎

Depends on

Used by

Every compact uniform space is complete Lemma

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. Wodzicki, Uniform Structure (standard reference, not scraped)