I learned a cute proof that [0, 1] is compact today, and felt the need to share it. By some basic point-set topology, this implies that [0, 1]^n is compact for any n, and then this implies that any closed and bounded subset of is compact; the converse is easy, so this proves the Heine-Borel theorem.
To prove it, we will use the infinite Ramsey theorem, which can be stated as “For every party with guests, either are all mutual friends, or are all mutual strangers.” There are various elementary or not-so-elementary proofs of the infinite Ramsey theorem; see Wikipedia’s article on Ramsey’s theorem for an example. It is a vast generalization of the result from elementary combinatorics that says that “For every party with 6 guests, either 3 are friends or 3 are strangers.” Here is the countable infinite cardinal.
We now claim that every sequence of real numbers has a monotone subsequence. This completes the proof, since any bounded monotone sequence is Cauchy.
Say that are friends if ; otherwise, say they are strangers. By the infinite Ramsey theorem, there is an infinite set such that either all elements of A are friends, or all elements of A are strangers. Passing to the subsequence indexed by A, we find a subsequence consisting only of friends (in which case it is increasing) or of strangers (in which case it either has a subsequence which is constant, or a subsequence which is decreasing).
By the way, this “friendly” property of is quite special. If is an uncountable cardinal such that for every party with guests, either are mutual friends or are mutual strangers, then we say that is “weakly compact”, and can easily prove that is a large cardinal in the sense that it is inaccessible and has a higher strength than an inaccessible cardinal (or even of a proper class of them).
some compact thoughts 