proof of Thm 21.7

Compact - Thm 21.7

Proof 1 → 2

For compact set $K\subseteq \mathbb{R}$,

Bounded

we can prove it’s bounded at first by:
Consider $H_m=(-m,m)$ as an open set, then according to

$$K\subseteq \bigcup _{m=1}^{\infty }{H}_m=\mathbb{R}$$

we can say $H_m$ is an subcover of $K$. According to $K$ is compact, we can say $(H_m)$ is finite. Then we have

$$K\subseteq \bigcup _{m=1}^M{H}_m=(-M,M)$$

Hence, we can say $K$ is bounded by $(-M,M)$.
$\square$

Closed

Proof 3 → 2