I'm working through the countable subadditivity of outer measure proof in Axler (Measure, Integration & Real Analysis), and I'm finding it difficult to understand the purpose or intuition of the rearrangement of sequences.
Suppose $A_1$, $A_2$, $\ldots$ is a sequence of subsets of $\mathbb{R}$. Then $$\left\lvert\bigcup\limits_{k=1}^{\infty}A_k\right\rvert \leq \sum\limits_{k=1}^{\infty}\left\lvert A_k \right\rvert.$$
For each $k \in \mathbb{Z}^+$, let $I_{1, k}, I_{2, k}, \ldots$ be a sequence of open intervals whose union contains $A_k$ such that
$$\sum\limits_{j=1}^{\infty} l(I_{j, k}) \leq \frac{\epsilon}{2^k} + \left\lvert A_k\right\rvert. $$ Thus $$\sum\limits_{k=1}^{\infty}\sum\limits_{j=1}^{\infty}l(I_{j, k}) \leq \epsilon + \sum\limits_{k=1}^{\infty} \left\lvert A_k \right \rvert.$$
Most proofs I see (or read) just replace the inner sum with the previous result, and use the "sequence" of open intervals $\{I_{1, 1}, I_{2, 1}, \ldots\}, \{I_{1, 2}, I_{2, 2}, \ldots\}$ (i.e. k = 1, k = 2 etc.), claiming that their union contains $\bigcup_{k=1}^{\infty} A_k$. Axler defines a new sequence $$\{I_{1, 1}\}, \{I_{1, 2}, I_{2, 1}\}, \{I_{1, 3}, I_{2, 2}, I_{3, 1}\}, \ldots$$ and I've seen derivations as to why the sum of lengths of these intervals are the same as that of the double summation, but I don't understand why the sequence needs to be constructed in the first place. I would really appreciate some light being shed on that.
