LemmaStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31

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.

Subnets preserve eventual properties and every limit of a net

Statement

If $y$ is a subnet of a net $x$ , then every subset in which $x$ is eventually contained is one in which $y$ is eventually contained. Consequently every limit of $x$ is a limit of $y$ .

Facts & Assumptions

Given: A subnet $y_e=x_{\phi(e)}$ of $x_d$ .

[A1]

Eventual cofinality says that for every $d_0\in D$ some $e_0\in E$ has $e\ge e_0\Rightarrow\phi(e)\ge d_0$ (Subnet via an eventually cofinal index map).

[A2]

A net converges to $p$ exactly when it is eventually in every neighbourhood of $p$ (Convergence and cluster points of a net in a topological space).

Proof

technique · direct

1.1

Suppose $x$ is eventually in $S\subseteq X$ , and choose $d_0$ such that $d\ge d_0$ implies $x_d\in S$ .

given

2.1

Choose $e_0$ from [A1] for this $d_0$ ; then $e\ge e_0$ gives $y_e=x_{\phi(e)}\in S$ . Thus $y$ is eventually in $S$ .

step 1.1A1

3.1

If $x$ converges to $p$ , apply step 2.1 to each neighbourhood of $p$ using [A2]; then $y$ is eventually in every such neighbourhood and converges to $p$ .

step 2.1A2∎

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: 8 results over 5 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

Schlumprecht, Math 655 notes (standard reference, not scraped)
Net (mathematics) (Wikipedia) (standard reference, not scraped)
Subnet (mathematics) (Wikipedia) (standard reference, not scraped)