sequentially compact

for common Metric Space, compact $⟺$ sequentially compact

Def 20.1

$S\subset (X,d)$ is called sequentially compact if every sequence has a convergence subsequence with limit in $S$

Lemma 20.2

for compact sequence $S$
$T\subseteq S$ is closed $⟹$ $T$ is sequentially compact

Thm 20.3 21.1

$S\subseteq {\mathbb{R}}^d$ and sequence $S$ compact or sequentially compact $⟺$

1. $S$ is bounded
2. $S$ is closed