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

Heine-Borel theorem for the line (Tao)

Let $X$ be a subset of $R$ . Then the following two statements are equivalent:

[...].
Given any sequence $(a_{n})_{n = 0}^{\infty}$ of real numbers which takes values in $X$ (i.e., $a_{n} \in X for all n$ ), 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).

09.03.2020