Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 2 / 25J / e / Solution

Let $U$ be one uniform random variable on $(0,1)$, and defineFor every $U>0$ the sequence is eventually zero, so $X_n\to 0$ almost surely. But $\mathrm{Pr}(|X_n|>1/2)=1/n$, whose sum diverges. This sequence is not completely convergent; the exceedance events are nested and dependent.