Show that a subset of $\mathbb{R}$ is compact iff it is closed and bounded.
By definition of compact, a set $S\subset\mathbb{R}$ is compact if every open covering of $S$ has a finite subcovering. So for a given open covering {$U_{n}$}, there exista a finite covering $U_1,...,U_n$ of {$U_{n}$} s.t. $S\subset U_1\cup...\cup U_n$.
Now that I have all this out of the way, I'm not exactly sure where to start.