If $(f_n)$ is a sequence in $L(X,\mathbb{X}, \mu)$ which converges uniformly on $X$

Question

If $(f_n)$ is a sequence in $L(X,\mathbb{X}, \mu)$ which converges uniformly on $X$ to a function $f$, and if $\mu (X)< \infty$, then

$\int fd\mu=\lim \int f_n d \mu$

Any suggestion? I'm trying to get a function $g$ that limits $|f_n|$, to use the dominated convergence theorem, but without success.

Answer 1

$|\int (f_n-f) \, d\mu| \leq \int |f_n-f| \, d\mu <\epsilon \mu (X)$ for $n$ sufficiently large.