= Solution
<Convergence in probability> means $\mathbb P(|X_n-X|>\epsilon)\to0$ for every $\epsilon>0$. <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 $|X_n-X|>\epsilon$ give
$$
F_X(t-\epsilon)-\mathbb P(|X_n-X|>\epsilon)\leq F_{X_n}(t)\leq F_X(t+\epsilon)+\mathbb P(|X_n-X|>\epsilon).
$$
Take lower and upper limits as $n\to\infty$, then $\epsilon\downarrow0$ at a continuity point. Both bounds tend to $F_X(t)$, proving that \b[convergence in probability implies convergence in distribution].
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