Is the following statement true? Convergence in probability implies almost sure convergence

No. The typewriter sequence on [0,1] P(Xₙ ≠ 0) → 0 but every ω is hit infinitely often, so there is no a.s. limit.

Counterexample

Is this true?

Convergence in probability implies almost sure convergence

No — it is false.

The counterexample

The typewriter sequence on [0,1]

Why it fails

$P(X_{n} \ne 0) \to 0$ but every $\omega$ is hit infinitely often, so there is no a.s. limit.

Keep practicing

Limit assumed to exist

Reasoning with a limit before establishing there is one. limsup is not lim.

Published 23 Aug 2026. Found a mistake? Tell us — corrections are the fastest thing we act on.