Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31
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  • 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.

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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\mathbb N and its identity sequence xn=nx_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 xx is a composite x∘hx\circ h with h:N→Nh:\mathbb N\to\mathbb N strictly increasing; such an hh is injective (A strictly increasing index map satisfies nk≥kn_k \ge k).

Refutation

technique · direct
1.1

Put ϕ(0)=0\phi(0)=0 and ϕ(k)=k−1\phi(k)=k-1 for k≥1k\ge1, and let yk=xϕ(k)y_k=x_{\phi(k)}. For every nn, all k≥n+1k\ge n+1 satisfy ϕ(k)=k−1≥n\phi(k)=k-1\ge n, so ϕ\phi is eventually cofinal and yy is a subnet of xx.

A1
2.1

The subnet has y0=y1=0y_0=y_1=0. Every subsequence of the injective identity sequence is injective by [A2], so yy cannot be a subsequence of xx.

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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 76 results over 20 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