The Lebesgue outer measure for $E\cup F\subset \mathbb{R}^d$ is defined as$$m^*(E\cup F)=\inf_{\cup_{n=1}^\infty B_n\supset (E\cup F), B_n\text{ boxes}}\sum_{n=1}^\infty|B_n|$$
in Terence Tao's measure theory book, page 23. In the proof for finite subadditivity for separated sets he lets $\varepsilon > 0$, and asserts that we can cover $E\cup F$ by a countable family of boxes $B_1, B_2, \cdots$ such that
$$\sum_{n=1}^\infty|B_n|\leq m^*(E\cup F)+\varepsilon$$
I don't see why this should be true for any $\varepsilon > 0$; by the definition we obviously have
$$m^*(E\cup F)\leq\sum_{n=1}^\infty|B_n|$$
because we were taking the infimum of that sum in the definition of outer measure. Why does adding any $\varepsilon > 0$ result in the reverse inequality in this step of the proof?