Prove or disprove $\lim_{n \to +\infty}\mathbb{P}(X_n \in B) = \mathbb{P}(X\in B)$

Question

Let $\{X_n : n\geq 1\}$ and X be random variables om some probability space with probability density $f_n$ and $f$ respectively.

If $f_n \to f$ almost everywhere then for every $B \in \mathscr{B}(\mathbb{R})$ ( $\mathscr{B}$ for Borel set), prove or disprove $\lim_{n \to +\infty}\mathbb{P}(X_n \in B) = \mathbb{P}(X \in B)$.

I appreciate it if you can tell me if the statement holds so that I can work on a proof or a counterexample.

Thanks

Answer 1

This statement is true. If you want an approach, I leave one in a comment.