Suppose $F_n \to F $ weakly, $ x \in c(F)$ and $ x_n $ is a real sequence converging to x. Prove that $F_n(x_n) \to F(x) $. Here $F_n$, $F$ are cdfs and $c(F)$-set of continuity points of $F$.
I proceeded this way:
Since $F$ is continuous $\forall x_n, x \in c(F)$,
$F(x_n) \to F(x)\quad\forall x_n, x \in c(F)$.
So $\exists N_1 \in \mathscr{N}$ such that $\forall n \ge N_1$, $|F(x_n)-F(x)|\le \frac {\varepsilon}{2}$
Again since $F_n \to F $ weakly, $\exists N_2 \in \mathscr{N}$ such that $\forall n \ge N_2$, $|F_n(x)-F(x)|\le \frac {\varepsilon}{2}$
Therefore $|F_n(x_n)-F(x)|\le |F_n(x_n)-F(x_n)| + |F(x_n)-F(x)| \le \varepsilon $ $\forall n \ge \max(N_1,N_2)$ and $x_n,x \in c(F)$
Now what if $x_n$ does not belong to $c(F)$? How to prove the result in that case?