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apalmer3
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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?