I would like to see other examples of (ideas of) proofs and results in the same vein as the proof of the Ax-Grothendieck theorem. To explain what I mean by "in the same vein", I will quote from the corresponding wikipedia article:

This method of proof is noteworthy in that it is an example of the idea that finitistic algebraic relations in fields of characteristic $0$ translate into algebraic relations over finite fields with large characteristic.Thus, one can use the arithmetic of finite fields to prove a statement about $\mathbb{C}$ even though there is no non-trivial homomorphism from any finite field to $\mathbb{C}$.The proof thus uses model theoretic principles to prove an elementary statement about polynomials.The proof for the general case uses a similar method.

Edit(after the comment by Angelo to Martin's answer below): answers along the line of "using finite fields for problems concerning infinite fields" are also welcome.