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

By Tonelli theorem for the nonnegative indicators,If this is finite, then $N$ is finite almost surely: alternatively, Markov inequality givesSince $\{N=\infty \}\subseteq \{N\ge m\}$ for every $m$,This is the first Borel-Cantelli lemma.