$\int_E fd\mu=\lim \limits_{n\to\infty}\int_E f_nd\mu$

Question

Let $(X,\mathcal S, \mu)$ be a measure space, $E\in \mathcal S$, $\mu(E)<\infty$ and $f_n$ be a non negative sequence, measurable real-valued functions that converge uniformly to $f:X\rightarrow \mathbb R$. How can I prove $\int_E f d\mu=\lim_\limits {n\to\infty}\int_E f_nd\mu?$ I tried to use Lebesgue's dominated convergence theorem but I didn't get any further here because I need a function $g$ with $|f_n|\leq g$ for all $n$