Let $ f\in L^{1}(E) $ and $ \{E_n\}$ be a sequence of measurable subsets of $E$. If $$ \lim_{n\to +\infty} m(E_n) = 0$$
prove that $$ \lim_{n\to +\infty} \int_{E_n} f = 0.$$
I tried to interchange the integration and the limit with a theorem such as DCT, but I failed to find a suitable function. I would appreciate a hint!
Hint: Given $\epsilon>0$ there is a simple function $\phi$ with $\int|f-\phi|<\epsilon$. And now since $\phi$ is simple it is bounded...