Let be $\mu:S\rightarrow[0, +\infty]$ positive measure on $S \ \sigma-algebra$ on $X$ such that $\mu(X)<+\infty$, and a $f_n:X\rightarrow\mathbb{K} \ (n \geq 1)$ a sequence of $S-measurable$ functions, and $f:X\rightarrow\mathbb{K}$ a function such that: $$\forall n\geq1, \ f_n \ is \ bounded$$ $$f_n \ is \ uniformly \ convergent \ to \ f \ on \ X \ for \ n\rightarrow+\infty$$ Prove that $f_n$ and $f$ are $\mu-integrable \ (\int_X|f|d\mu, \int_X|f_n|d\mu<+\infty$) on $X$ and prove that: $$\lim_{n\rightarrow+\infty}\int_X|f_n-f|d\mu=0$$ also prove that this is false if $\mu(X) = +\infty$.
So, I was able to prove that $f_n$ is $\mu-integrable$, and I know I have to use the dominated convergence theorem, but I don't know how to find an upperbound $\mu-integrable$ function $g$ to use the theorem. Any help?