Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 3 / 24J / i / Solution

Convergence in probability means $\mathbb{P}(|X_n-X|>ϵ)\to 0$ for every $ϵ>0$. Convergence in distribution means $F_{X_n}(t)\to F_X(t)$ at every continuity point $t$ of $F_X$. The inclusions obtained by separating the event $|X_n-X|>ϵ$ giveTake lower and upper limits as $n\to \infty$, then $ϵ\downarrow 0$ at a continuity point. Both bounds tend to $F_X(t)$, proving that convergence in probability implies convergence in distribution.