CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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.

A divergent probability sum does not force infinitely many occurrences without independence

Statement refuted

If $(A_{n})_{n \in N}$ is a sequence of events with $\sum_{n = 0}^{\infty} P (A_{n}) = + \infty,$ then $P (A_{n} i . o .) = 1.$

Facts & Assumptions

Given: An event $A$ with $0 < P (A) < 1$ , and define $A_{n} : = A$ for every $n \in N$ .

[L1]

The event $A_{n} i . o .$ is the event that infinitely many of the $A_{n}$ occur. (Limsup and the infinitely often event)

[L2]

Probability measures respect complements and monotone set identities. (Basic identities for a probability measure)

Counterexample

technique · constructive

1.1

Since every $A_{n}$ equals $A$ , one has $\sum_{n = 0}^{\infty} P (A_{n}) = \sum_{n = 0}^{\infty} P (A) = + \infty .$

givenalgebraconstruct

1.2

For every outcome $ω$ , either $ω \in A$ and then $ω \in A_{n}$ for all $n$ , or $ω \notin A$ and then $ω \notin A_{n}$ for all $n$ . Hence the infinitely-often event is exactly $A$ itself: $A_{n} i . o . = A .$

2.1

Therefore $P (A_{n} i . o .) = P (A),$ and the chosen hypothesis $0 < P (A) < 1$ makes this probability strictly between $0$ and $1$ . So the displayed implication is false without an independence hypothesis.

step 1.1step 1.2L2discharge-construct∎

Depends on

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Dependency tree · two levels

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