I'm trying to prove the following claim:
$f_n \in C_c$, $C_c$ being the set of continuous functions with compact support, then $\mathrm{lim}_{n \rightarrow \infty} || f_n - f||_{\infty} = 0$ implies $f_n(x) \rightarrow f(x)$ uniformly.
So, according to my understanding,
$ \mathrm{lim}_{n \rightarrow \infty} || f_n - f||_{\infty} = 0 $ $ \Leftrightarrow $ $ \forall \varepsilon > 0 \exists N: n > N \Rightarrow |f_n(x) - f(x)| \leq || f_n - f||_{\infty} < \varepsilon$ $\mu$-almost everywhere on $X$.
Now my problem is that I don't see how uniform convergence follows from pointwise convergence $\mu$ almost everywhere. Can someone give me a hint? I guess I have to use the fact that they have compact support but I don't see how to put this together.
Thanks for your help!