Let us recall the following theorem.
Theorem. Let $(X,\| \cdot \|)$ be a normed vector space. The space $X$ is complete (w.r.t. the metric induced by the norm) if and only if every totally convergent series is convergent in norm.
A series $\sum_n x_n$ is totally convergent when $\sum_n \|x_n\|$ converges (as a sum of non-negative real numbers).
Apply this result to $X=L^1$. Your question turns out to be equivalent to (or implied by) the completeness of $L^1$ (w.r.t its natural norm). And how do you prove that $L^1$ is complete? Exactly by choosing any Cauchy sequence and constructing a subsequence that is almost everywhere convergent. This is the reason why (1) is proved before (2).