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

For $\epsilon >0$,using $1-x\le e^{-x}$. The final probabilities are summable, so Borel--Cantelli implies that eventuallyIntersecting over positive rational $\epsilon$ proves the second extremes of independent exponential variables assertion