What are the most elegant proofs that you have learned from MO? - MathOverflow

What are the most elegant proofs that you have learned from MO?

2010-06-27T00:11:58Z

One of the things that MO does best is provide clear, concise answers to specific mathematical questions. I have picked up ideas from areas of mathematics I normally wouldn't touch, simply because someone posted an eye-catching answer on MO.

In particular, there have been some really elegant and surprising proofs. For example, this one by villemoes, when the questioner asked for a simple proof that there are uncountably many permutations of $\mathbb{N}$.

The fact that any conditionally convergent series [and that such exists] can be rearranged to converge to any given real number x proves that there is an injection P from the reals to the permutations of $\mathbb{N}$.

Or this one by André Henriques, when the questioner asked whether the Cantor set is the zero set of a continuous function:

The continuous function is very easy to construct: it's the distance to the closed set.

There must many such proofs that most of us have missed, so I'd like to see a list, an MO Greatest Hits if you will. Please include a link to the answer, so that the author gets credit (and maybe a few more rep points), but also copy the proof, as it would nice to see the proofs without having to move away from the page.

(If anyone knows the best way to copy text with preservation of LaTeX, please advise.)

I realize that one person's surprise may be another person's old hat, so that's why I'm asking for proofs that you learned from MO. You don't have to guarantee that the proof is original.

Answer by Joel David Hamkins for What are the most elegant proofs that you have learned from MO?

2010-06-27T02:05:34Z

In this fantastic answer, Ashutosh proved that the Axiom of Choice is equivalent to the assertion that every set admits a group structure.

In ZF, the following are equivalent:

(a) For every nonempty set there is a binary operation making it a group

(b) Axiom of choice

Non trivial direction [(a) -> (b)]:

The trick is Hartogs construction which gives for every set $X$ an ordinal $\aleph(X)$ such that there is no injection from $\aleph(X)$ into $X$. Assume for simplicity that $X$ has no ordinals. Let $o$ be a group operation on $X \cup \aleph(X)$. Now for any $x \in X$ there must be an $\alpha \in \aleph(X)$ such that $x o \alpha \in \aleph(X)$ since otherwise we get an injection of $\aleph(X)$ into $X$. Using $o$, therefore, one may inject $X$ into $(\aleph(X))^{2}$ by sending $x \in X$ to the <-least pair $(\alpha, \beta)$ in $(\aleph(X))^{2}$ such that $x o \alpha = \beta$. Here, < is the lexic well ordering on the product $(\aleph(X))^{2}$. This induces a well ordering on $X$.

(The argument is due originally to Hajnal and Kertész, 1973.)

Answer by Eric O. Korman for What are the most elegant proofs that you have learned from MO?

2010-06-27T02:06:37Z

Unfortunately I can't find the link but someone mentioned this proof that there are irrational numbers $a$ and $b$ such that $a^b$ is rational: if $\sqrt{2}^\sqrt{2}$ is rational then we are done, if it is irrational then $2 = (\sqrt{2}^\sqrt{2})^\sqrt{2}$ is an irrational raised to an irrational.

Answer by Daniel Barter for What are the most elegant proofs that you have learned from MO?

2010-06-27T03:57:21Z

I asked a question a while ago about proving that the real line is connected.

http://mathoverflow.net/questions/26537/connectedness-and-the-real-line

Omar Antolín-Camarena's Answer and comment prove that the closed interval $[0,1]$ is connected iff it is compact.

Answer by Ostap Chervak for What are the most elegant proofs that you have learned from MO?

2011-04-15T15:37:09Z

I found several very nice proofs which I enjoyed:

1.Brilliant proof of fundamental theorem of algebra by Gian Maria Dall'Ara http://mathoverflow.net/questions/10535/ways-to-prove-the-fundamental-theorem-of-algebra/10684#10684

2.Some proofs of quadratic reciprocity: http://mathoverflow.net/questions/1420/whats-the-best-proof-of-quadratic-reciprocity (I especially liked that one: http://mathoverflow.net/questions/1420/whats-the-best-proof-of-quadratic-reciprocity/1431#1431)

3.Proof that $\mathbb{R}^{2n+1}$ does NOT have a square root (quite elementary and beatiful) http://mathoverflow.net/questions/60375/is-r3-the-square-of-some-topological-space/60389#60389

4.Nullstellensatz using model theory http://mathoverflow.net/questions/9667/what-are-some-results-in-mathematics-that-have-snappy-proofs-using-model-theory/9693#9693

5.If in ring R every countably generated ideal is principal than R is a PID http://mathoverflow.net/questions/8042/do-there-exist-non-pids-in-which-every-countably-generated-ideal-is-principal/8067#8067

6.An infinite dimensional vector space have smaller dimension than it's dual. http://mathoverflow.net/questions/13322/slick-proof-a-vector-space-has-the-same-dimension-as-its-dual-if-and-only-if-it/13372#13372

7.Topological proof that Z is a Bezout domain. http://mathoverflow.net/questions/42512/awfully-sophisticated-proof-for-simple-facts/64039#640397.

Answer by François G. Dorais for What are the most elegant proofs that you have learned from MO?

2011-07-24T20:08:59Z

My favorite is this proof by Bjorn Poonen that every finite Galois extension of $\mathbb{Q}$ has infinitely many completely split primes. Although Bjorn's proof does not give the density of such primes, as the proof using the Chebotarev Density Theorem does, it is refreshing to see that such an elementary proof exists.

Answer by Ali Enayat for What are the most elegant proofs that you have learned from MO?

2011-07-25T03:13:50Z

My candidate is Jim Belk's one-line answer to the question about the existence of functions from $\Bbb{R}$ to $\Bbb{R}$ whose range is $\Bbb{R}$ on every open interval.

I do wonder, however, if Jim Balk's solution was known to founders of classical set theory (Cantor, Bernstein, Hausdorff, ...).