Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passverified 2026-08-09 (gpt-5.6-terra-codex-subscription)
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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.

Every convergent filter on a uniform space is Cauchy

Statement

Every filter converging in the induced topology of a uniform space is Cauchy.

Facts & Assumptions

Given: A filter F\mathcal F converging to xx in a uniform space.

[L1]

Filter convergence means that every neighbourhood of the limit belongs to the filter (Convergence and cluster points of a filter on a topological space).

[L2]

Entourage balls form neighbourhood bases, and every entourage has a symmetric square root (The sets containing an entourage ball about each of their points form a topology, Uniform space in the entourage formulation, Every uniformity has a base of symmetric entourages).

[L3]

A Cauchy filter has an EE-small member for every entourage EE (Cauchy filter in a uniform space).

Proof

technique · direct
1.1

Let EE be an entourage and choose a symmetric entourage DD with D∘D⊆ED\circ D\subseteq E.

L2choose
1.2

The neighbourhood D[x]D[x] belongs to F\mathcal F by convergence.

L1L2
2.1

Since DD is symmetric, D[x]×D[x]⊆D−1∘D=D∘D⊆ED[x]\times D[x]\subseteq D^{-1}\circ D=D\circ D\subseteq E, so D[x]D[x] is EE-small.

step 1.1step 1.2
3.1

As EE was arbitrary, F\mathcal F is Cauchy by [L3].

step 2.1L3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 20 results over 12 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources