So one of the axioms of outer measure is the following:

Countable subadditivity axiom: If $\displaystyle A = \bigcup_{n=1}^{\infty} A_n $ then $\displaystyle m^{*}A \leq \sum_{n=1}^{\infty} m^{*}A_n $.

Well it's not really an axiom since we want/have to prove it. But what is the point of using the "$\displaystyle \varepsilon/2^n $ trick" to prove this? E.g. we have the following:

$\displaystyle \sum_{k=1}^{\infty} |I_{kn}| < m^{*} A_n+ \frac{\varepsilon}{2^n} $ (1)

This is analogous to $\displaystyle \inf S \leq x \Rightarrow x < \inf S + \varepsilon $ for some $\displaystyle \varepsilon >0 $ (with $\displaystyle x \in S $). But what is special about $\displaystyle \varepsilon/2^n $? Why not just use $\displaystyle \varepsilon $?