Why is the expected value of Kolmogorov Smirnov statistic converges to 0?

Question

Let $X_1,...,X_n$ be a random sample with distribution function $F$ and let $F_n$ be the corresponding empirical distribution function:

$$F_n(x) = \frac{1}{n} \sum_{i=1}^{n}1_{(-\infty,x]}(X_i)$$

My book says that $\lim_{n->\infty}E(\sup_{x\in R}|F_n(x) - F(x)|) = 0$ can be proved? How do I do this using Glivenko Cantelli?

Answer 1

The combination of Glivenko Cantelli with the dominated convergence theorem works, since $\lvert F_n(x)-F(x)\rvert\leqslant 2$.