# Convergence in Distribution and Convergent Series

probability theory

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.

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 $$x' s.t. $$|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.$$