In my ongoing love affair with compactness I am constantly revisiting a particular proof of the Heine-Borel theorem, a characterization of compactness in . There are two proofs that I know of: the standard “subdivision” proof and the “creeping along” proof. I am going to focus on the creeping along proof.

Heine-Borel Theorem.A subset is compact if and only if it is closed and bounded.

To do some creeping we need to collect some useful facts.

**Fact 1.** A subset is bounded if and only if is contained in some closed interval

**Fact 2.** The set is complete (as a linear order) because every non-empty set with an upper bound has a least upper bound, called .

**Fact 3.** Closed subsets of compact subsets of are in fact themselves compact. With fact 1 this means that it is enough to show that closed and bounded *intervals* in are compact. (In general closed subsets of compact spaces are compact.)

So now let us creep: