Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07
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.

Subsequence characterization of convergence in probability

Statement

Xn→X in probability if and only if every subsequence of (Xn) has a further subsequence converging almost surely to X.

Facts & Assumptions

Given: Real random variables (Xn) and X on one probability space.

[L1]

Almost-sure convergence implies convergence in probability (Almost-sure convergence implies convergence in probability).

[L2]

Probability convergence has an almost-surely convergent subsequence (An almost-surely convergent subsequence from convergence in probability).

Proof

technique · direct
1.1

If Xn→X in probability, every subsequence has the same property. Apply [L2] to that subsequence to obtain the asserted further subsequence.

L2
2.1

Conversely, if probability convergence failed, some ε>0 and a subsequence would satisfy P(∣Xnk−X∣>ε)≥ε for every k. Any almost-surely convergent further subsequence would converge in probability by [L1], a contradiction.

L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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