Question on proof that closed subset of compact metric space is compact 
Proposition. If $K$ is a compact metric space, $F \subset K$, and $F$ is closed, then $F$ is compact.
Proof. Suppose $F$ is a closed subset of the compact set $K$. If $\mathcal{G} = \{G_\alpha\}$ is an open cover of $F$, then $\mathcal{G}' = \mathcal{G} \cup \{F^c\}$ will be an open cover of $K$. We are given that $K$ is compact, so let $\mathcal{H}$ be a finite subcover of $\mathcal{G}'$ for $K$. If $F^c$ is one of the open sets in $\mathcal{H}$, omit it. The resulting finite subcover of $\mathcal{G}$ will be an open cover of $F$.

I have a question. So I understand that we can throw out $F^c$, as it's disjoint from $F$. But how do we know that a finite subcover of $\mathcal{G}' = \mathcal{G} \cup \{F^c\}$ (which for all purposes and intents, just consider $\mathcal{G}$) for $K$ is actually an open cover of $F$? How do we know we aren't missing any part of $F$, i.e. there's a part of $F$ that hasn't been covered?

This follows from the fact $F \subset K$.
In other words, if $\mathcal{G}$ is an open cover for $K$, then $$K \subset \bigcup_{\alpha\in A} G_{\alpha} $$
and thus $\mathcal{G}$ must also be an open cover for $F$, since $$F \subset K, \quad K \subset \bigcup_{\alpha\in A} G_{\alpha} $$ $$\implies F \subset \bigcup_{\alpha\in A} G_{\alpha}$$ In other words, subset inclusion is transitive.
