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Which are the deepest theorems with the most elementary proofs?
I give two examples:
i) Proof_of_the_Euler_product_formula_for_the_Riemann_zeta_function
ii) Proof that the halting problem is undecidable using diagonalization

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closed as not a real question by Amitesh Datta, Asaf Karagila, Pete L. Clark, Raskolnikov, Jonas Teuwen Oct 23 '11 at 18:37

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.

This should be community wiki, I think. – Grumpy Parsnip Oct 22 '11 at 2:33
Isn't depth, by definition, inversely proportional to the elementariness of proofs? – Henning Makholm Oct 22 '11 at 2:43
How about "reopen, please"?!? Also "PLZ reopen" in the title is obnoxious. – The Chaz 2.0 Oct 26 '11 at 14:23
@GM2001: Please stop editing titles to contain messages like "AWESOME" or " REOPEN !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!...please!!!!!!!!!!‌​!!!!!!!!!!!!!!!!!!!!!". This is not the first time you're doing it. – ShreevatsaR Oct 26 '11 at 14:36
@GM2001: If you edit this question any more, I will lock it, which will prevent further edits, comments, and answers. – Zev Chonoles Oct 26 '11 at 14:37

These perhaps aren't particularly deep, but they are the first that come to mind.

  1. Irrationality of $\sqrt{2}$ by contradiction.
  2. Uncountability of the reals by diagonalization.
  3. Existence of graphs with arbitrarily high girth and chromatic number by the probabilistic method.
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+1 Because I hadn't been aware of #3 before. – anon Oct 22 '11 at 2:38
I think the linked proof in (1) is misnamed, and is not a proof by infinite descent (unlike this one)‌​. – r.e.s. Oct 22 '11 at 3:22

I think one should not confuse "important with "deep". The facts that $\sqrt{2}$ is irrational, that there is no surjective map $X\to2^X$, or that there are an infinity of primes, are certainly important or even "fundamental", but their proofs are so simple that one cannot call them "deep". A theorem is "deep" when its proof is really hard and, above all, requires a theory that transcends the realm the problem is formulated in. Consider, e.g., Gauss' theorem about which regular $n$-gons can be constructed with ruler and compass.

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I think the theorem that $\sqrt{2}$ is irrational would have met your criterion for depth at the time it was discovered. Imagine that the Greeks saw $\sqrt{2}$ as the length of the diagonal of a unit square (rather than as the positive solution to $x^2=2$). Then to show the irrationality of $\sqrt{2}$, we need to transcend geometry to go to number theory, where we have available the fundamental theorem of arithmetic. – Srivatsan Oct 23 '11 at 15:49
To add on Srivatsan's comment, the proof of $|P(X)|>|X|$ while seemingly trivial nowadays required the development of an entire new field in mathematics. I'd say this qualifies as pretty deep. – Asaf Karagila Oct 23 '11 at 19:35

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