$F_n \overset{w}{\to} F$, and $F$ is continous. Show that $F_n$ converges to $F$ uniformly on $\mathbb{R}$

Question

$\{F_n\}$ and $F$ are distribution functions, and $F$ is continuous on $\mathbb{R}$. If $F_n$ converges weakly to $F$, show that $$ \sup_x | F_n(x) - F(x) | \to 0, n \to \infty $$

I know that weak convergence is equivalent to the condition that $F_n$ converges to $F$ at every continuity of $F$, but how to show the uniform convergence here? Much appreciation to any help!

Answer 1

We can use the following strategy:

• first show that if $t_n\to t$, then $$\lim_{n\to\infty}|F_n(t_n)-F_n(t)|=0.$$ You will need to use the weak convergence of $F_n(t)$ to $F(t)$ and the fact that $F_n$ is non-decreasing.
• second, fix a positive $\varepsilon$, reduce the uniform convergence on the real line to a uniform convergence on a compact interval and argue by contradiction.