Heine-Borel_Theorem

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Definition

A set $K \subseteq \mathbb{R}$ is compact if and only if it is closed and bounded.

Proof

($\Rightarrow$) Assume $K$ is compact.

Bounded: The open cover $\{(-n, n) : n \in \mathbb{N}\}$ covers $K$. By compactness, a finite subcover exists, so $K \subseteq (-N, N)$ for some $N$.

Closed: Let $x$ be a limit point of $K$. For each $n$, there exists $x_n \in K$ with $|x_n - x| < 1/n$. The sequence $(x_n)$ has a convergent subsequence (by compactness) converging to some $y \in K$. But $(x_n) \to x$, so $x = y \in K$.

($\Leftarrow$) Assume $K$ is closed and bounded. Let $(x_n)$ be a sequence in $K$. By Bolzano-Weierstrass_Theorem, $(x_n)$ has a convergent subsequence. Since $K$ is closed, the limit is in $K$. Thus $K$ is compact.