Alphabeta Math
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.

Every cluster point of a universal net is a limit of that net

Statement

Every cluster point of a universal net is a limit of that net.

Facts & Assumptions

Given: A universal net x:D→Xx:D\to X and a cluster point pp.

[A1]

A universal net is eventually in SS or eventually in X∖SX\setminus S for every subset SS (Universal net: eventually in every subset or eventually in its complement).

[A2]

Clusterhood is frequent membership in every neighbourhood, while convergence is eventual membership in every neighbourhood (Convergence and cluster points of a net in a topological space).

Proof

technique · contradiction
1.1

Assume for a contradiction that xx does not converge to pp. Then some neighbourhood NN of pp is not an eventual set for xx.

A2assume-contra
2.1

By universality, xx is eventually in X∖NX\setminus N. This contradicts frequent membership in NN, since an index after both thresholds would lie in N∩(X∖N)N\cap(X\setminus N).

step 1.1A1A2
3.1

Therefore xx converges to pp.

step 2.1discharge-contradiction∎

Depends on

Used by

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