Past exam of the probability course of the University of Oxford / 2021 / 1 / b / iii / Solution

Use $X_n\sim \mathrm{Unif}[0,1/n]$ from part (a). We already proved convergence in distribution to zero. For any prescribed $p\in [0,1]$, put $c_n=p/n$. Then $c_n\to 0$, and the cumulative distribution function satisfiesThis includes $p=0$ and $p=1$, since the uniform distribution assigns zero probability to each endpoint. The limit cumulative distribution function is discontinuous at zero, explaining why the conclusion of (ii) does not apply there.