Creeping Along

In my ongoing love affair with compactness I am constantly revisiting a particular proof of the Heine-Borel theorem, a characterization of compactness in \mathbb{R} . 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 A \subseteq \mathbb{R} 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 \mathcal{A} \subseteq \mathbb{R} is bounded if and only if is contained in some closed interval [a,b]

Fact 2. The set \mathbb{R} is complete (as a linear order) because every non-empty set A \subseteq \mathbb{R} with an upper bound has a least upper bound, called \sup A .

Fact 3. Closed subsets of compact subsets of \mathbb{R} are in fact themselves compact. With fact 1 this means that it is enough to show that closed and bounded intervals in \mathbb{R} are compact. (In general closed subsets of compact T_2 spaces are compact.)

So now let us creep:

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