I wanna prove a portion of the Heine-Borel Theorem because I didn't like the proof I was given. I'm only going to be proving the *closed* part. I will **not** be proving the *bounded* part because that part was fine.

**Theorem** In $\mathbb{R}^n$ with the Euclidean topology, compact sets are precisely the closed and bounded subsets of $\mathbb{R}^n$

*Proof*

Let $K$ be a compact (bounded) subset of $\mathbb{R}^n$. Toward a contradiction, suppose $K$ is not closed.

Since $K$ is not closed, let $x\_o$ be an accumulation point of $K$ which is not in $K$.

Consider all $k \in K.$ Since $K$ is bounded, let $M =$ max$\{|k\_1|,|k\_2|,|k\_3|,...\}$. Then for all $k \in K,$ $-M \leq k \leq M,$ bounding all of $K$.

Consider $\mathcal{U}\_i =\{k \in K: d(k,x\_o)<\frac{M}{i}\}$ for $i \in \mathbb{N}$

Then $\displaystyle\bigcup\_{i=1}^\infty \mathcal{U}\_i$ is an open cover of $K$ with no finite subcover

This contradicts our assumption that $K$ was not closed.

Thus, the compact set $K$ is closed.