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## Homework Statement

Prove directly (i.e., without using the Heine-Borel theorem) that if K [tex]\subseteq[/tex] Rd is compact and F [tex]\subseteq[/tex]K is closed, then F is compact.

## Homework Equations

Definition of a Compact Set: A set K is said to be compact if, whenever it is contained in the union of a collection G={Ga} of open sets, then it is also contained in the union of some finite number of the sets in G.

## The Attempt at a Solution

I think that I need to be more detailed in this, so any help would be greatly appreciated. :-D

If K is compact, then there exists a finite subcover for every open cover of K. If F [tex]\subseteq[/tex] K, then every open cover of K also covers F. And because K is compact, every open subcover of K is once again a cover for F.

Therefore F is compact.

It seems to me that there has to be more to this proof. Help?