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Grothendieck's proof of the well-known Ax–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?

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closed as not a real question by quid, unknown (google), Bill Johnson, Martin Brandenburg, Fernando Muro Jul 8 '12 at 18:21

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

Too vague and thus too broad. Vote to close. (BTW, please use CW mode when asking for a list of answers; there is a box to tick, you can still do this by editing the question.) – quid Jul 8 '12 at 15:00
Maybe you can look at here:… – Mahdi Majidi-Zolbanin Jul 8 '12 at 15:16