Let $X_{n}, n = 1, 2...$ be i.i.d. random variables, with continuous cumulative distribution function $F$. Let
$F_{n}(x) = \frac{\sum_{i = 1}^{n} \chi_{X_{i} \leq x}(\omega)}{n}$.
Show $||F_{n} - F ||_{\infty} \xrightarrow{} 0$ a.s.
We have that $F_{n} \xrightarrow{} F$ a.s., by the strong law of large numbers.
Also, could anyone please share how you would deal with the infinity norm when you saw it? Even it's definition alone is kinda complicated. I tried to write this question using the definition of $L^{\infty}$ norm but it didn't seem to help.