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What are the most important examples of theorems and definitions which are post factum obvious, i.e., hard to put together but easy to understand and use (and prove, in the case of theorems) once you see them?

So far the best examples of this sort that have come to mind are the Schur lemma (although perhaps I am misunderstanding the level of difficulty of its original proof -- please correct me if I am wrong on this one) and, to a somewhat limited extent, the notions of category and functor, but I am sure there are many more, and I would be delighted to learn about them.

I am interested both in the cases when the true significance of such results was understood immediately and when it took a while. Let me also stress that the present question is not quite the same as Examples of mathematical results discovered "late" inter alia because it also includes the definitions.

Full disclosure: this question is motivated by my academia SE question on the best ways to convey the significance and value of such results when writing a paper or giving a talk.

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  • $\begingroup$ $0.999...=1$ might be an example since so many people find it weird. But then again, it might more be a matter of notation $\endgroup$ – Alice Ryhl Sep 22 '14 at 17:46
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    $\begingroup$ One is reminded of a joke: "According to the Nobel Prize-winning physicist Richard Feynman, mathematicians designate any theorem as "trivial" once a proof has been obtained--no matter how difficult the theorem was to prove in the first place. There are therefore exactly two types of true mathematical propositions: trivial ones, and those which have not yet been proven." $\endgroup$ – Zhen Lin Sep 22 '14 at 18:00
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    $\begingroup$ Maybe something like ${n \choose k} = {n-1 \choose k} + {n-1 \choose k-1}$? Once you think of the argument, it's simple and obvious, but thinking of the argument isn't entirely obvious if you're new to combinatorics. I still think this question is too vague, though. $\endgroup$ – Jack M Sep 23 '14 at 21:32
  • $\begingroup$ @JackM: Why do you think it's vague? I am strongly convinced that such results do exist... $\endgroup$ – just-learning Sep 23 '14 at 21:44

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