Past exam of the mathematics course of the University of Cambridge / 2022 / ia / Paper 2 / 12F / c

If ${\sum }_n\mathbb{P}(A_n)=\infty$, part (b) givesThus $2^{-N}=0$ almost surely. By the convention in the question, this occurs exactly when $N=\infty$. ThereforeThis is the second Borel-Cantelli lemma for independent events.