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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).