A “friendly” proof of the Heine-Borel theorem

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 \mathbb R^n 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 \omega guests, either \omega are all mutual friends, or \omega 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 \omega is the countable infinite cardinal.

We now claim that every sequence (x_n)_n of real numbers has a monotone subsequence. This completes the proof, since any bounded monotone sequence is Cauchy.

Say that n < m are friends if x_n < x_m; otherwise, say they are strangers. By the infinite Ramsey theorem, there is an infinite set A \subseteq \mathbb N 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 \omega is quite special. If \kappa is an uncountable cardinal such that for every party with \kappa guests, either \kappa are mutual friends or \kappa are mutual strangers, then we say that \kappa is “weakly compact”, and can easily prove that \kappa 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).


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