There are two notions of compactness, they turn out to be equivalent for metric spaces.

Let $X$ be a metric space, $K \In X$ a subset.

• sequential compactness: we say $K$ is compact, if every sequence in $K$ has a convergent subseq.
• compactness: any open cover of $K$ admits a finite subcover.

The two notions turns out are equivalent, see https://courses.wikinana.org/math104-f21/compactness

We will follow Pugh to give a proof. See also Rudin Thm 2.41

We will finish discussion about compactness. In particular, in $\R^n$, we have Heine-Borel theorem, namely, compact subsets are exactly those subsets which are closed and bounded. Note that, this is false for general metric space, e.g. closed and bounded subset in $\Q$ are not compact.