Let $(E,\cal{E})$ be a measurable space, and let $f:E\rightarrow\mathbb{R}$ be an elementary function, i.e. $f=\sum_{i=1}^{\infty}a_i\, I_{A_i}$ where for each $i\geq 1$, $ \,I_{A_i}$ is the indicator function of $A_i$, the $A_i$ are disjoint, and $A_i \in \cal{E}$ for each $i$.
One way to prove this is discussed here: Elementary functions are measurable
My question: could this be proved as a limit of simple functions?
i.e., let $f_n = \sum_{i=1}^{n} a_i I_{A_i}$. We know that each $f_n$ is simple, and thus measurable on $\cal{E}$. And we know the limit of measurable functions is measurable.
I feel like i'm missing something. Is this a valid proof?