If $F_n\overset{d}{\longrightarrow}F$, $\{x_n\}_{n\ge1}$ is any sequence of reals and $F$ is continuous then, $F_n(x_n)-F(x_n)\rightarrow0$.
From Polya's theorem, we know that since the continuity condition is satisfied,
$$\sup_{x\in\mathbb{R}}|F_n(x)-F(x)|\rightarrow0.$$
Fix $\epsilon>0$. There exists $n_0$ s.t. for all $n\ge n_0$, $\sup_x|F_n(x)-F(x)|<\epsilon\Leftrightarrow F_n(x)-F(x)<\epsilon\forall x$.
We have, for all $n\ge n_0$, $F_n(x_n)-F(x_n)<\epsilon$
$\Rightarrow F_n(x_n)-F(x_n)\rightarrow0$
Is this proof okay? Or are there steps I am missing?