Beautiful theorems with short proof [closed]

I like to ask for beautiful mathematical theorems with short proof. A proof is short in my sense if it is at most one page assuming basic notations and very basic results a second year student will know and understand. I do not like to discuss what beauty means...I have now written down a script with 107 such theorems (in my sense) and their proof and I have ideas for another 20. One the one hand I would be very thankful for theorems I have yet not considered. On the other hand I like to see if there is some consensus which theorems with a short proof are beautiful.

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This should be big list and community wiki... – Igor Rivin Sep 10 at 23:56
I believe many results here qualify: papers.assafrinot.com/short.pdf – Andres Caicedo Sep 11 at 0:03
Several of the "one-page papers" that I put up some years ago at math.harvard.edu/~elkies/Misc/index.html#papers should fit the bill. – Noam D. Elkies Sep 11 at 1:28
I've flagged moderators to make CW. – Benjamin Steinberg Sep 11 at 2:03
I've CWed.​​​​​ – Anton Geraschenko Sep 11 at 2:19

closed as subjective and argumentative by Felipe Voloch, Henry Cohn, Benjamin Steinberg, Todd Trimble, S. SraSep 11 at 5:27

The standard evaluation of $\int_{-\infty}^{\infty} \exp(-x^2) dx.$

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 Sure! This is very nice. – Jörg Neunhäuserer Sep 11 at 11:38

Cantor's diagonal argument to prove that $\mathbb{R}$ is uncountable.

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Picard's little theorem is remarkable, and its one-line proof led Littlewood to remark that it would be the world's shortest Ph.D. thesis.

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 It is indeed a very slick proof, but the original question says: "A proof is short in my sense if it is at most one page assuming basic notations and very basic results a second year student will know and understand." How much complex analysis should these students know? – Yemon Choi Sep 11 at 6:17

L.M. Kelly's proof of the Sylvester-Gallai theorem: in any configuration of $n$ points in the plane, not all on a line, there is a line containing exactly two of the points.

See Aigner & Ziegler, "Proofs from the Book", chapter 8.

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While you are at it, see the rest of Proofs from The Book. – Gerald Edgar Sep 11 at 1:47
I think about including Sylvester-Gallai theorem from the book of proofs. But many proves in the book are to long. – Jörg Neunhäuserer Sep 11 at 11:31

Fermat's proof, by infinite descent, that there is no Pythagorean right triangle whose area is a square might qualify.

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 Many thanks, thats one at have not considerd! – Jörg Neunhäuserer Sep 11 at 11:34

@Paul Monsky's proof of Monsky's theorem: a complete proof starting from nothing takes two pages. (doesn't quite meet the criteria, but what the heck).

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 Thanks for the nice link. – Jörg Neunhäuserer Sep 11 at 11:34

Euler's formula $$\mathrm{e}^{i \theta} = \cos \left( \theta \right) + i \sin \left( \theta \right)$$ when considered as a theorem. From whatever angle you look at it, almost all the proofs are short and extremely beautiful.

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As someone who is teaching a course in the spirit of "foundations of analysis" -- short? really? – Yemon Choi Sep 11 at 6:15
Hi Yemon. Assuming only basic notations and very basic results, which proofs did you have in mind that are longer than one page? – an12 Sep 11 at 6:54
Dear Yemon, i think the same. – Jörg Neunhäuserer Sep 11 at 11:33
@an12: how are you defining cos and sin and exp? – Yemon Choi Sep 12 at 21:36

The proof(via the pigeon-hole principle--continued fractions would need too much preparation) that when D>0 is not a square then the "Pellian equation" xx-Dyy=1 has a non-trivial solution.

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 Thanks, agian one I have not considered. – Jörg Neunhäuserer Sep 11 at 11:36

The proof of Brouwer fixed point theorem by using fundamental group of $S^1$ is equal to $\mathbb{Z}$, while the fundamental group of $D^2$ is trivial.

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This actually requires much more than a page of background. – Igor Rivin Sep 11 at 3:51
I agree with Igor – Jörg Neunhäuserer Sep 11 at 11:35