CounterexampleConstruction: Literature-sourcedVerification: 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.

Convergence in probability need not be almost sure

Statement refuted

Convergence in probability need not imply almost-sure convergence.

Facts & Assumptions

Given: Independent events $E_{n}$ with $P (E_{n}) = 1 / (n + 1)$ , and $X_{n} = 1_{E_{n}}$ .

[L1]

Pairwise independent events with divergent probability sum occur infinitely often almost surely (Second Borel-Cantelli lemma under pairwise independence).

[L2]

Convergence in probability is fixed-threshold tail convergence (Convergence in probability).

Counterexample

technique · direct

1.1

For $0 < ε < 1$ , $P (∣ X_{n} ∣ > ε) = 1 / (n + 1) \to 0$ ; for $ε \geq 1$ it is zero. Hence $X_{n} \to 0$ in probability by [L2].

2.1

But $\sum_{n} P (E_{n}) = \infty$ , so [L1] gives $E_{n}$ infinitely often almost surely. On that event $X_{n}$ has infinitely many values $1$ , and therefore cannot converge to $0$ .

step 1.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

S. Roch, Lecture 3: Modes of convergence, Example 3.11 (standard reference, not scraped)