I just learned about Furstenberg's proof of the infinitude of primes (link goes to Wikipedia article); it was published in 1955 in the American Mathematical Monthly. (Link goes to JSTOR. Full citation: The American Mathematical Monthly, Vol. 62, No. 5. (May, 1955), p. 353. If you have to do any work to get the article from JSTOR -- for example, logging in through a proxy server because you're at home instead of on campus -- or you don't have JSTOR access -- don't worry, you're not missing much. The article's a third of a page, and the Wikipedia article is probably longer than Furstenberg's original.) Surprisingly, I hadn't seen this before.
Anyway, here's my version of the proof: topologize the integers by taking the (doubly infinite) arithmetic sequences as a basis. (The only thing that needs checking here, to show these form a basis, is that the nonempty intersection of two basis elements contains a basis element; this is true since the intersection of two arithmetic sequences is another arithmetic sequence.) Consider the set which consists of all the integers except -1 and 1; this is the union of the sequences pZ over all primes. Now, there are no nonempty finite open sets in this topology, so there are no cofinite closed sets except Z itself. Thus the union of the pZ isn't closed. So it can't be an union of finitely many closed sets, since finite unions of closed sets are closed. So there are infinitely many sets pZ, and thus infinitely many primes!
(Thanks to an anonymous commentator for pointing out some errors.)