Dense subset of $L^1(X)$

Let $(X,\mu)$ be a $\sigma$-finite mearsure space. Suppose $E_1\subset E_2\subset \cdots$ is a sequence of subsets with finite measure such that $X= \cup E_i$.

Let $A = \{ f\in L^1(X): f \text{ is supported in some } E_n\}$. Then I want to show that A is dense in $L^1(X)$.

My idea is to show that every characteristic function $\chi_E$ can be approximated by $A$, $E$ being a finite mearsurable subset. But I need to show that $E$ is contained in some $E_n$, almostly.

I am not sure the last statement is true or not. Can somebody give a hint, thanks very much.