Let $\displaystyle K \subset R^1$ consist of $\displaystyle 0$ and the numbers $\displaystyle 1/n$ for any $\displaystyle n = 1, 2, 3 ...$ Prove that $\displaystyle K$ is compact directly from the definition (w/o using the Heine-Borel Theorem).

Definition from the book:

A subset $\displaystyle K$ of a metric space $\displaystyle X$ is said to be compact if every open cover of $\displaystyle K$ contains a finite subcover. More explicitly, the requirement is that if $\displaystyle \{G_{\alpha}\}$ is an open cover of $\displaystyle K$, then there are finitely many indices $\displaystyle \alpha_1, ... \alpha_n$ such that $\displaystyle K \subset G_{\alpha_1} \bigcup ... \bigcup G_{\alpha_n}.$