What are the most elegant proofs that you have learned from MO? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T17:13:16Z http://mathoverflow.net/feeds/question/29652 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/29652/what-are-the-most-elegant-proofs-that-you-have-learned-from-mo What are the most elegant proofs that you have learned from MO? John Stillwell 2010-06-27T00:11:58Z 2011-07-25T03:13:50Z <p>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.</p> <p>In particular, there have been some really elegant and surprising proofs. For example, <a href="http://mathoverflow.net/questions/29475/an-easy-proof-of-the-uncountability-of-bijections-on-natural-numbers/29488#29488" rel="nofollow">this one</a> by villemoes, when the questioner asked for a simple proof that there are uncountably many permutations of $\mathbb{N}$.</p> <blockquote> <p>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 $\mathbb{N}$.</p> </blockquote> <p>Or <a href="http://mathoverflow.net/questions/24034/can-cantor-set-be-the-zero-set-of-a-continuous-function/24041#24041" rel="nofollow">this one</a> by André Henriques, when the questioner asked whether the Cantor set is the zero set of a continuous function:</p> <blockquote> <p>The continuous function is very easy to construct: it's the distance to the closed set.</p> </blockquote> <p>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.</p> <p>(If anyone knows the best way to copy text with preservation of LaTeX, please advise.)</p> <p>I realize that one person's surprise may be another person's old hat, so that's why I'm asking for proofs that <em>you learned</em> from MO. You don't have to guarantee that the proof is original.</p> http://mathoverflow.net/questions/29652/what-are-the-most-elegant-proofs-that-you-have-learned-from-mo/29662#29662 Answer by Joel David Hamkins for What are the most elegant proofs that you have learned from MO? Joel David Hamkins 2010-06-27T02:05:34Z 2010-06-27T02:05:34Z <p>In <a href="http://mathoverflow.net/questions/12973/does-every-non-empty-set-admit-a-group-structure-in-zf/12988#12988" rel="nofollow">this fantastic answer</a>, Ashutosh proved that the Axiom of Choice is equivalent to the assertion that every set admits a group structure. </p> <blockquote> <blockquote> <p>In ZF, the following are equivalent:</p> <p>(a) For every nonempty set there is a binary operation making it a group</p> <p>(b) Axiom of choice</p> <p>Non trivial direction [(a) -> (b)]:</p> <p>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 &lt;-least pair $(\alpha, \beta)$ in $(\aleph(X))^{2}$ such that $x o \alpha = \beta$. Here, &lt; is the lexic well ordering on the product $(\aleph(X))^{2}$. This induces a well ordering on $X$.</p> </blockquote> </blockquote> <p>(The argument is due originally to Hajnal and Kertész, 1973.)</p> http://mathoverflow.net/questions/29652/what-are-the-most-elegant-proofs-that-you-have-learned-from-mo/29663#29663 Answer by Eric O. Korman for What are the most elegant proofs that you have learned from MO? Eric O. Korman 2010-06-27T02:06:37Z 2010-06-27T02:06:37Z <p>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.</p> http://mathoverflow.net/questions/29652/what-are-the-most-elegant-proofs-that-you-have-learned-from-mo/29674#29674 Answer by Daniel Barter for What are the most elegant proofs that you have learned from MO? Daniel Barter 2010-06-27T03:57:21Z 2010-06-27T03:57:21Z <p>I asked a question a while ago about proving that the real line is connected. </p> <p><a href="http://mathoverflow.net/questions/26537/connectedness-and-the-real-line" rel="nofollow">http://mathoverflow.net/questions/26537/connectedness-and-the-real-line</a></p> <p>Omar Antolín-Camarena's Answer and comment prove that the closed interval $[0,1]$ is connected iff it is compact. </p> http://mathoverflow.net/questions/29652/what-are-the-most-elegant-proofs-that-you-have-learned-from-mo/61833#61833 Answer by Ostap Chervak for What are the most elegant proofs that you have learned from MO? Ostap Chervak 2011-04-15T15:37:09Z 2011-06-05T11:10:47Z <p>I found several very nice proofs which I enjoyed:</p> <p>1.Brilliant proof of fundamental theorem of algebra by Gian Maria Dall'Ara <a href="http://mathoverflow.net/questions/10535/ways-to-prove-the-fundamental-theorem-of-algebra/10684#10684" rel="nofollow">http://mathoverflow.net/questions/10535/ways-to-prove-the-fundamental-theorem-of-algebra/10684#10684</a></p> <p>2.Some proofs of quadratic reciprocity: <a href="http://mathoverflow.net/questions/1420/whats-the-best-proof-of-quadratic-reciprocity" rel="nofollow">http://mathoverflow.net/questions/1420/whats-the-best-proof-of-quadratic-reciprocity</a> (I especially liked that one: <a href="http://mathoverflow.net/questions/1420/whats-the-best-proof-of-quadratic-reciprocity/1431#1431" rel="nofollow">http://mathoverflow.net/questions/1420/whats-the-best-proof-of-quadratic-reciprocity/1431#1431</a>)</p> <p>3.Proof that $\mathbb{R}^{2n+1}$ does NOT have a square root (quite elementary and beatiful) <a href="http://mathoverflow.net/questions/60375/is-r3-the-square-of-some-topological-space/60389#60389" rel="nofollow">http://mathoverflow.net/questions/60375/is-r3-the-square-of-some-topological-space/60389#60389</a></p> <p>4.Nullstellensatz using model theory <a href="http://mathoverflow.net/questions/9667/what-are-some-results-in-mathematics-that-have-snappy-proofs-using-model-theory/9693#9693" rel="nofollow">http://mathoverflow.net/questions/9667/what-are-some-results-in-mathematics-that-have-snappy-proofs-using-model-theory/9693#9693</a></p> <p>5.If in ring R every countably generated ideal is principal than R is a PID <a href="http://mathoverflow.net/questions/8042/do-there-exist-non-pids-in-which-every-countably-generated-ideal-is-principal/8067#8067" rel="nofollow">http://mathoverflow.net/questions/8042/do-there-exist-non-pids-in-which-every-countably-generated-ideal-is-principal/8067#8067</a> </p> <p>6.An infinite dimensional vector space have smaller dimension than it's dual. <a href="http://mathoverflow.net/questions/13322/slick-proof-a-vector-space-has-the-same-dimension-as-its-dual-if-and-only-if-it/13372#13372" rel="nofollow">http://mathoverflow.net/questions/13322/slick-proof-a-vector-space-has-the-same-dimension-as-its-dual-if-and-only-if-it/13372#13372</a></p> <p>7.Topological proof that Z is a Bezout domain. <a href="http://mathoverflow.net/questions/42512/awfully-sophisticated-proof-for-simple-facts/64039#640397" rel="nofollow">http://mathoverflow.net/questions/42512/awfully-sophisticated-proof-for-simple-facts/64039#640397</a>.</p> http://mathoverflow.net/questions/29652/what-are-the-most-elegant-proofs-that-you-have-learned-from-mo/71150#71150 Answer by François G. Dorais for What are the most elegant proofs that you have learned from MO? François G. Dorais 2011-07-24T20:08:59Z 2011-07-24T20:08:59Z <p>My favorite is this proof by Bjorn Poonen that <a href="http://mathoverflow.net/questions/15220/is-there-an-elementary-proof-of-the-infinitude-of-completely-split-primes/15221#15221" rel="nofollow">every finite Galois extension of $\mathbb{Q}$ has infinitely many completely split primes</a>. 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.</p> http://mathoverflow.net/questions/29652/what-are-the-most-elegant-proofs-that-you-have-learned-from-mo/71176#71176 Answer by Ali Enayat for What are the most elegant proofs that you have learned from MO? Ali Enayat 2011-07-25T03:13:50Z 2011-07-25T03:13:50Z <p>My candidate is <a href="http://mathoverflow.net/questions/32126/function-with-range-equal-to-whole-reals-on-every-open-set/32128#32128" rel="nofollow">Jim Belk's one-line answer</a> to the <a href="http://mathoverflow.net/questions/32126/function-with-range-equal-to-whole-reals-on-every-open-set/32127#32127" rel="nofollow">question</a> about the existence of functions from $\Bbb{R}$ to $\Bbb{R}$ whose range is $\Bbb{R}$ on every open interval. </p> <p>I do wonder, however, if Jim Balk's solution was known to founders of classical set theory (Cantor, Bernstein, Hausdorff, ...).</p>