Integral when limit of nondecreasing simple functions greater than another simple function

Question

This looks intuitive but I have a problem writing the proof.

Let $(f_n)$ (nondecreasing) and $f$ all be nonnegative simple functions in a measure space. If $\lim f_n\geq f$ pointwise, then $\lim\int f_n\geq \int f$.

Here $\int f:=\sum^{n}_{k=1}a_k m(A_k)$ when $f=\sum^{n}_{k=1}a_k\chi_{A_k}$ is the simple representation.

I tried: (1) to assume $\lim f_n>f$ so that $f_m>f$ for some $m,m+1,...$ and use monotonicity of integral, but what about equality? Or (2) (maybe just) obtaining $\int(\lim f_n)=\lim(\int f_n)$ but I don't know how as I am so noob in analysis.

Thanks a lot for any suggestion.