Because , wouldn't it ?
So one of the axioms of outer measure is the following:
Countable subadditivity axiom: If then .
Well it's not really an axiom since we want/have to prove it. But what is the point of using the " trick" to prove this? E.g. we have the following:
This is analogous to for some (with ). But what is special about ? Why not just use ?
As you're dealing with a limit, you want to appear. This one is arbitrary !
Since you'll sum from 1 to infinity, if you keep , you'll sum something positive an infinite amount of times.
Which is irrelevant for your problem.
is arbitrary and > 0. It's a good candidate, because of this infinite sum.
And you'll get in the end.
You could also have taken or whatever you want. The main point is to get a converging series, arbitrary small.