For the closed part, I just noted that it's a compact subspace of a Hausdorff space and therefore it's closed. For the bounded part, I know intuitively that since every open cover has a finite subcover, I just have to take the largest ball that includes all these covers, but I don't know how to write it rigorously. I know they'd all fit a large ball...
I also must find a metric space in which not every closed subspace is compact. Which is an example of a metric space in which not every closed is compact? Because once I know that, I could just take the metric $0,1$
An insightful cover to use:
Let $S$ be the compact set. Pick any point $x \in S$. Now consider the cover $\{B(x, n) \ | \ n \in \mathbb{N} \}$, where $B(x, n)$ denotes the open ball centered at $x$ of radius $n$. Notice that $S$ is contained inside the largest $B(x, k)$ in the finite subcover this cover admits.