I'm asked to prove that $\{0,1, 1/2, 1/3, \dots \}$ is a compact set without the help of Heine-Borel Theorem. I have to use only the definition of a compact set.
$\textbf{My work}$:
As long as the definition of a compact set says that $K$ is compact if for every open covering of $K$ there is a finite subcovering (is this the right word in English?) of $K$, then I started taking an arbitrary open covering, say $\mathcal{G} = \{G_{\lambda}: \lambda \in L\}$. Well, now I have to prove that there is a finite subcovering. If $A = \{0,1, 1/2, 1/3, \dots \} \subseteq \bigcup G_{\lambda}$, then every element $a$ of $A$ is such that $a \in G_{\lambda}$ for some $\lambda$. But I'm stuck at this point. Any ideas/hints? Thanks!
Best Answer
Hint: at least one open set of the cover contains $0$. How many elements of the set are not covered by this one open set? Can you cover the rest of the set using finitely many more open sets?