# 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).