I was trying to understand completely the post of Terrence Tao on Ax-Grothendieck theorem.

This is very cute. Using finite fields you prove that every injective polynomial map $\mathbb C^n\to \mathbb C^n$ is bijective. It seems to me that the only point in the proof presented in the post that is not explained completely is the following lemma:

Lemma. Take any finitely generated ring over $\mathbb Z$ and quotient it by a maximal ideal. Then the quotient is a finite field.

Is there some comprehensible reference for the proof of this lemma?

In slightly different wording, the question is the following: assuming Nullstellensatz, can one really give a complete proof of Ax-Grothendick theorem in two pages, so that it can be completely explained in one (2 hours) lecture of an undergraduate course on algebraic geometry?