17 - Heine-Borel

Heine-Borel

Let $X$ be a subset of ${\mathbb{R}}^n$

$$X\text{ is compact}⟺X\text{ is closed and bounded}$$

Proof

Proof $⟹$
Apply

1. Compactness implies sequentially compactness
2. Closed and boundedness of sequentially compact subspaces

Proof $⟸$
Apply

1. Bolzano-weierstrass
2. ^6eefad