Prove that if F is closed subset of a compact set K in R then F is compact bu using the definition below......

the def. of compact set: for every open cover of the set it has a finite subcover.....
hint : start with:
Let G :={G_a : a in some interval} be an open cover of F to show it has a finite subcover
Let the open cover is G U -F (U is the union)

I want the solution in details .................
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