Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 27L / a / Solution

The statements and proof are in Borel-Cantelli lemmas. For the first,
$\mathbb{P}({\bigcup }_{n\ge m}{A}_n)\le {\sum }_{n\ge m}\mathbb{P}(A_n)\to 0$. For independent events in the divergent case,and then let $m\to \infty$.