I don't understand how a compact set can have a finite open subcover given certain infinite open covers.
For example, say my set $K\subset \mathbb{R}^n$ is the closure of some ball around a point: $K=\overline{B_r(p)}$. Since we're in Euclidean space and $K$ is closed and bounded, $K$ is compact. Consider the sequence of open sets $\{\overline{B_{r/n}(p)}^c\}_{n=1}^\infty$. Their infinite union is an open cover of $K$ but any finite union isn't; namely, any finite union doesn't contain $p$.
Best Answer
$p$ isn't in $\cup_{n=1}^\infty \overline{B_{r/n}(p)}^c,$ so it's not an open cover:
$p=\cap_{n=1}^\infty \overline{B_{r/n}(p)}=\left(\cup_{n=1}^\infty \overline{B_{r/n}(p)}^c\right)^c$