Did you know that if R is a ring, M is a finite R-module, and φ : M —> M is a surjective module map, then φ is an isomorphism? Just learned this today. This is Lemma 4.4a in Eisenbud if you want a reference. Or see Lemma Tag 05G8 in the stacks project.
If you know about limit arguments etc, then you immediately see how to prove it for finitely presented modules (reduce to Noetherian case, etc, etc). Thinking about it some more you may come to the conclusion that this is one of those things that is simply not true for finite modules in general. So I enjoy lemmas like this since it feels as if you are getting away with something!
[Edit: Just (6:02 PM) received an alternative proof of this lemma from Thanos D. Papaïoannou which is I would say a more honest and in particular completely standard proof. So now there are two proofs… More anybody?]