Prove the Countable additivity of Lebesgue Integral.
You can affirm that (there is no "n" in the member of the right in your question, makes confusion):
The $E_n$ are disjoint sets, so $$\forall N \in \mathbb{N}, \ \sum_{n=1}^N \int_{E_n} f=\int_{\bigcup\limits_{n=1}^N E_n} f = \int_{A_N}f = \int_E f_N$$
The you take the limit when $N$ goes to $\infty$, and i think you are done!