Prove that if $f \in L^{1}(A)$ and {${A_{n}}$} is a sequence of measurable subsets of A with $\lim_{n \to \infty}m(A_{n})=0$ then $lim_{n \to \infty} \int_{A_n}f=0$
I know there exists a very similar post for tackling this particular problem but I cannot find the complete proof of it. I would be grateful if someone could help.