Let $X_n, n \in \mathbb{N}$ and $X$ be random variables with $F_n$ and $F$ being the respective CDF's. Additionally we have that $F_n \overset{D}{\rightarrow}F$ and $x_n \rightarrow x_0, x_n, x_0 \in \mathbb{R}$, for $n \rightarrow \infty$ and $\mathbb{P}(X = x_0) = 0$ then $\lim_{n \rightarrow \infty} F_n(x_n) = F(x_0)$.
At first I was thinking that we could apply the MCT and exchange the limits, but i am really at a loss here.
Best Answer
Since $F$ is continuous at $x_0$, for each $\epsilon>0$ there exists $\delta>0$ s.t. $|F(x)-F(x_0)|<\epsilon$ whenever $|x-x_0|<\delta$. Fix $\epsilon>0$ and chose $x'<x_0<x''$ s.t. $|x'-x_0|\vee |x''-x_0|<\delta$ and $x'$ and $x''$ are continuity points of $F$ (we can do that because the set of continuity points of $F$ is dense in $\mathbb{R}$). Then, since $x_n$ is eventually between $x'$ and $x''$, $$ F(x')= \lim_{n\to\infty}F_n(x')\le \liminf_{n\to\infty}F_n(x_n)\le \limsup_{n\to\infty}F_n(x_n)\le \lim_{n\to\infty}F_n(x'')=F(x''), $$ and, therefore, $$ F(x_0)-\epsilon\le \liminf_{n\to\infty}F_n(x_n)\le \limsup_{n\to\infty}F_n(x_n)\le F(x_0)+\epsilon. $$