TITLE: Open Cover / Real Analysis
QUESTION [6 upvotes]: I have the next question: Let $K \subset $ $R^1$ consist of $0$ and the numbers 1/$n$, for $n=1,2,3,\ldots$ Prove that $K$ is compact directly from the definition (without using Heine-Borel).
I'm trying to understand compact sets so I would be grateful if someone could give me some examples of open covers and subcovers. Thank you!

REPLY [2 votes]: Let $\{U_\alpha\}_{\alpha \in A}$ be a cover of $K$.  Then there exists some $\alpha_0 \in A$ such that $U_{\alpha_0}$ contains $0$.  Since $U_{\alpha_0}$ is open, for some $\epsilon > 0$, we have an open ball $B(0,\epsilon)$ centered at $0$ such that $B(0,\epsilon) \subset U_{\alpha_0}$.  Since $\epsilon >  0$, we can find $n \in \mathbb{N}$ such that $\epsilon > 1/n$.  Thus, $B(0,\epsilon)$ contains $\{0,1/n, 1/(n+1), \ldots\}$.  Now take elements of the open cover containing $1/2, \dots, 1/(n-1)$.  Since only finitely many elements are not in $U_{\alpha_0}$, we only need finitely many open sets in our cover.