Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 27G / b / i / Solution

Fix $\epsilon >0$. Since a rate-one exponential variable has
$\mathbb{P}(X_n>x)=e^{-x}$,The first Borel--Cantelli lemma gives the upper bound
$\mathrm{limsup}{X}_n/\log n\le 1$. Conversely,These events are independent, so the second lemma makes them occur infinitely often. Hence the lower bound is $1-\epsilon$. Letting rational $\epsilon \downarrow 0$ proves