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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. [...].
  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).