In the proof of the following proposition: Let $\{E_n\}$ be a countable collection of sets of real numbers. Then $ m^\ast\left(\bigcup E_n\right)\leq \sum m^\ast\left(E_n\right)~,$
we suppose that $m^\ast(E_n)$ is finite for all $n$. Then for each $E_n$, there is a countable collection $\{I_{k}^{n}:k\geq 1\}$ such that $E_n\subset \bigcup_{k}I_{k}^{n}$ $~~$ and $\sum l\left(I^{n}_{k}\right) \leq m^\ast(E_n)+\frac{\epsilon}{2^n},~~~\epsilon >0.$
Here is my question. I don't understand why the last inequality true. Explanations will be very much appreciated.
Thanks.