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## Homework Statement

Show that if E is a closed subset of a compact set F, then E is also compact.

## Homework Equations

I'm pretty sure you refer back to the Heine-Borel theorem to do this.

"A subset of E of R

^{k}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