Show that if E is a closed subset of a compact set F, then E is also compact.
I'm pretty sure you refer back to the Heine-Borel theorem to do this.
"A subset of E of Rk is compact iff it is closed and bounded"
The Attempt at a Solution
We are deal with metric spaces here. It should seem that I need to prove the same thing as in the second half of the Heine-Borel theorem. My textbook is proving Heine-Borel in a confusing way without clear statement/reason steps that I can apply to my problem.
-From the given information, E is in F
-From the given information, every open cover of F has a finite subcover of F
-From the given information, S \ E is open
...not sure where this leave me