One classical method is to show that if some property holds in infinite object, same or related thing must also hold for some (or all) large enough finite objec

One classical method is to show that if some property holds in infinite object, same or related thing must also hold for some (or all) large enough finite objects. Consider for example Grothendieck-Ax theorem: if f: C^n -> C^n is an injective (that is, one-to-one) multiple variable polynomial function over complex numbers, then it's also surjective. One of the ways to prove it is to show via model theory magic that if there was a counterexample to this theorem, there would also be a counterexample g: F^n -> F^n, where F is some finite field. But an injective function from finite set to finite set of the same size is also surjective.