Heine-Borel Theorem

This is compactness criterion for subsets of ${\mathbb{R}}^n$ .

Theorem

Let $A\subset {\mathbb{R}}^n$ with the standard metric. Then the following are equivalent:

1. $A$ is compact.
2. $A$ is closed and bounded.

Example

$$\left\{\frac{1}{n}|n\in \mathbb{N}\right\}\cup \{0\}$$

is compact. It is bounded by $(-1,1203201)$ and closed. Therefore by Heine-Borel it is compact in ${\mathbb{R}}^n$.

Lemma

Let $I_n=[a_m,b_m]\subset \mathbb{R}$ be closed and bounded $\forall m\in \mathbb{N}$. Assume $I_1\supset I_2\supset \cdots$. Then

$$\bigcap _{m\in \mathbb{N}}{I}_m=\varnothing$$

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Corollary (Infinite Subsets)

Since it is compact and closed, then every infinite subset has a limit point $p$ in it.