Understanding the proof of non-additivity of outer measure I am readying Axler's Measure, Integration and Real Analysis and I am stuck on a small portion of the following proof which is highlighted in yellow:
(2.19) suggests that for any $n$, $\displaystyle \bigcup_{k=1}^n (r_k+V) \leq 6$, no problem there. (2.20) is quite straightforward.
But I don't understand why (2.19) and (2.20) "suggest" that we choose (or can choose?) $n$ s.t. (2.21) is true.




I think it goes like this. You have established that $\left|V\right|>0$ so you can choose $n>6/\left|V\right|$ such that $\sum_{k=1}^n\left|\left(V+r_k\right)\right|=n\left|V\right|>6$, but then $\left|\cup^n_{k=1}\left(V+r_k\right)\right|\le 6$ for any $n$, so then you have:
$$
\left|\cup^n_{k=1}\left(V+r_k\right)\right|<\sum_{k=1}^n\left|\left(V+r_k\right)\right|
$$
i.e. with strict non-equality, which completes the proof
