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Ive proved the intersection of any number of closed sets is closed ...

and Let S = { A_a : a Element of I } be an collection of compact sets...then

by heine Borel Theorem ...Each A_a in S is closed...so this part is done now I just

have to show the intersection is bounded...

so I said since each A_a in S is bounded by Heine Borel ..then A_a is a subset of [a,b]

and then this is when Im really stuck:

Let B={[a,b] : There exists an A_a in S such that A_a subset of [a,b]}

I took the intersection of both sides of the subset ...and said Intersection S is a subset of

Intersection of [a,b] so Intersection of S is bounded and therefore compact

but somehow I think this is an error ..I dont know how to do this formally

Please help