Let $F_n\overset{d}{\longrightarrow}F$, then $\forall x\in\mathbb{R}$. Then, $$F(x-)\le\lim\inf F_n(x-)\le\lim\sup F_n(x)\le F(x)$$
I have the following:
Suppose $x_n\rightarrow x\in\mathbb{R}$. For large enough $n$, I have $$F(x-)\le F_n(x_n-)+\epsilon\le F_n(x_n)+\epsilon\le F(x)+\epsilon$$ But, I can't figure how to use this to prove the result which includes $F_n(x-)$ not $F_n(x_n-)$. Is it because $(x_n-)\rightarrow (x-)$ and $F_n$ is a CDF so, right continuous? If this is true, it would imply $\lim\inf F_n(x_n-)=\lim\inf F_n(x-)$ and I'll be done.