Proof that the Lebesgue measure is complete
It depends on what definition you use.
Using the Caratheodory criterion you can show that every set with outer measure zero is measurable.
Alternatively, a set $E$ is measurable, by definition, if for all $\varepsilon>0$ there exists an open set $U\supset E$ such that $m^*(U \setminus E)<\varepsilon$. Now this will easily imply completeness.