LemmaStatement: AI-adaptedProof: AI-generatedprecheck 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 $y_{e} = x_{ϕ (e)}$ of $x_{d}$ .

[A1]

Eventual cofinality says that for every $d_{0} \in D$ some $e_{0} \in E$ has $e \geq e_{0} \Rightarrow ϕ (e) \geq d_{0}$ (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 $d \geq d_{0}$ implies $x_{d} \in S$ .

given

2.1

Choose $e_{0}$ from [A1] for this $d_{0}$ ; then $e \geq e_{0}$ gives $y_{e} = x_{ϕ (e)} \in S$ . 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 · two levels

5 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

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