Let $(X, d)$ be a metric space, and $K$ an infinite subset of $X$. Show that $K$ is not compact in $(X, d)$.
From what I understand, a set of compact if every open cover of $K$ can be reduced to a finite subcover. Since $K$ is infinite, how can you construct a finite cover? Since you can't cover an "infinite amount of things" with a "finite amount of things," does this make $K$ not compact?
Or, should I take the "compactness $\implies$ closed and bounded" approach? Are all infinite sets not closed and unbounded (since I know $\mathbb{Z}$ is such in $\mathbb{R}$)?