Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 26H / b / Solution

Herefor every $0<\epsilon <1$, so $X_n\to 0$ in probability. On the other hand, the events $A_n=\{X_n=1\}$ are independent andThe second Borel-Cantelli lemmas givesThus $X_n$ cannot converge almost surely to zero. This independent rare-event counterexample to almost-sure convergence proves that convergence in probability need not imply almost-sure convergence of the full sequence.