I understand the proof up to the final step. It says that this holds $\forall \epsilon > 0$ and since $|N_{K|\mathbb{Q}}(a)|$ is always a positive integers, the existence of an $a \in \mathfrak{a}, a \neq 0$ with the wanted property follows.
For me it seems like it uses some kind of continuity. Maybe the norm is a continuous function (but what would the topology be?) and the ideal $\mathfrak{a}$ is closed. But I'm not sure whether this makes sense.
