Grothendieck's proof of the well-known Axâ€“Grothendieck Theorem uses the elegant argument that the existence of a counterexample for the field $\mathbb C$ imply the existence of a counterexample for a finite field. Roughly speaking, we transfer the problem from $\mathbb C$ to finite fields.

recently, Will Sawin, here in mathoverflow, proved that the ring of Hamilton quaternions over $\mathbb H_{2^s}$ is reversible proving that the existence of a counterexample fore some $s$ imply the existence of a counterexample for $s=1$ or $s=2$, (a finite problem tha is solved with a computer).

There are other examples of this type of proof?