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.

$\begingroup$ I believe many results here qualify: papers.assafrinot.com/short.pdf $\endgroup$– Andrés E. CaicedoSep 11, 2012 at 0:03

$\begingroup$ Zagier's paper "values of zeta functions and their applications" has a nice short proof of $\zeta(2)=\pi^2/6$ due to Calabi. $\endgroup$– PastenSep 11, 2012 at 0:08

$\begingroup$ Dear Caicedo, i am not sure but I belive these results are mostly not understandable for an average 2 second year student. $\endgroup$– Jörg NeunhäusererSep 11, 2012 at 0:13

1$\begingroup$ Several of the "onepage papers" that I put up some years ago at math.harvard.edu/~elkies/Misc/index.html#papers should fit the bill. $\endgroup$– Noam D. ElkiesSep 11, 2012 at 1:28

$\begingroup$ Dear Jörg, I cannot find your script on your homepage. Are you going to post it there? I would really like to see it! $\endgroup$– Piotr AchingerSep 11, 2012 at 2:31
8 Answers
The standard evaluation of $\int_{\infty}^{\infty} \exp(x^2) dx.$
Cantor's diagonal argument to prove that $\mathbb{R}$ is uncountable.

$\begingroup$ Improved by Cantor's first proof that the continuum is uncountable. Easier, too. $\endgroup$– AXHDec 7, 2013 at 23:14
Picard's little theorem is remarkable, and its oneline proof led Littlewood to remark that it would be the world's shortest Ph.D. thesis.

1$\begingroup$ 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? $\endgroup$ Sep 11, 2012 at 6:17
L.M. Kelly's proof of the SylvesterGallai 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.

3$\begingroup$ While you are at it, see the rest of Proofs from The Book. $\endgroup$ Sep 11, 2012 at 1:47

$\begingroup$ I think about including SylvesterGallai theorem from the book of proofs. But many proves in the book are to long. $\endgroup$ Sep 11, 2012 at 11:31
Fermat's proof, by infinite descent, that there is no Pythagorean right triangle whose area is a square might qualify.

$\begingroup$ Many thanks, thats one at have not considerd! $\endgroup$ Sep 11, 2012 at 11:34
The proof(via the pigeonhole principlecontinued fractions would need too much preparation) that when $D>0$ is not a square then the "Pellian equation" $x^2Dy^2=1$ has a nontrivial solution.
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.

$\begingroup$ As someone who is teaching a course in the spirit of "foundations of analysis"  short? really? $\endgroup$ Sep 11, 2012 at 6:15

$\begingroup$ Hi Yemon. Assuming only basic notations and very basic results, which proofs did you have in mind that are longer than one page? $\endgroup$– an12Sep 11, 2012 at 6:54

1

2$\begingroup$ @an12: how are you defining cos and sin and exp? $\endgroup$ Sep 12, 2012 at 21:36
@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).