False statementConstruction: AI-adaptedVerification: 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.

FALSE: every subnet of a sequence is a subsequence

Statement

FALSE. Every subnet of a sequence is a subsequence.

Facts & Assumptions

Given: The discrete topological space $N$ and its identity sequence $x_{n} = n$ .

[A1]

A subnet may use any eventually cofinal index map; it need not use a strictly increasing map (Subnet via an eventually cofinal index map).

[A2]

A subsequence of $x$ is a composite $x \circ h$ with $h : N \to N$ strictly increasing; such an $h$ is injective (A strictly increasing index map satisfies $n_{k} \geq k$ ).

Refutation

technique · direct

1.1

Put $ϕ (0) = 0$ and $ϕ (k) = k - 1$ for $k \geq 1$ , and let $y_{k} = x_{ϕ (k)}$ . For every $n$ , all $k \geq n + 1$ satisfy $ϕ (k) = k - 1 \geq n$ , so $ϕ$ is eventually cofinal and $y$ is a subnet of $x$ .

2.1

The subnet has $y_{0} = y_{1} = 0$ . Every subsequence of the injective identity sequence is injective by [A2], so $y$ cannot be a subsequence of $x$ .

step 1.1A2

3.1

Thus the stated universal claim is false.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

30 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)
Subnet (mathematics) (Wikipedia) (standard reference, not scraped)
Net (mathematics) (Wikipedia) (standard reference, not scraped)