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Math and science::Analysis::Tao::09. Continuous functions on R

Heine-Borel theorem for the line (Tao)

Let \( X \) be a subset of \( \mathbb{R} \). Then the following two statements are equivalent:

  1. \( X \) is closed and bounded.
  2. Given any sequence \( (a_n)_{n=0}^{\infty} \) of real numbers which takes values in \( X \) (i.e., \( a_n \in X \text{ for all } n\) ), there exists a subsequences \( (a_{n_j})_{j=0}^{\infty} \) of the original squence, which converges to some number \( L \) in \( X \).

Tao introduces Heine-Borel theorem quite separate to the Bolzano-Weierstrass theorem. I think that the Heine-Borel theory is more fitting to be grouped with the Bolzano-Weierstrass theorem; it is a theorem of sequences and not directly concerned with continuous functions (chapter 9, where it appears).

It's worth comparing this to the Bolzano-Weierstrass theorem. The Bolzano-Weierstrass theoem is (almost) enough to show the implication from a) to b) above. The main addition of the Hiene-Borel theorem is in the implication from b) to a). In fact, it requires the axiom of choice. (Well, Tao's proof uses it).

In the language of metric space topology, this is asserting that every subset of the real line which is closed and bounded is also compact.

There exists a more general version of this theorem, due to Eduard Heine (1821-1881) and Emile Borel (1871-1956).