Here's the problem: "Let C be a collection of closed sets of real numbers with the property that every finite subcollection of C has a nonempty intersection, and suppose that one of the sets is bounded. Then, [tex] \bigcap_{F \in C}^F \neq \emptyset [/tex]"(adsbygoogle = window.adsbygoogle || []).push({});

I've used the Heine-Borel theorem on this, so the bounded set is compact, ie has a finite open cover, etc etc, and De Morgan's law to get the intersection of a bunch of closed sets, but I don't know where to go next. I think Heine-Borel & DeMorgan's law is on the right track, but I'm not sure how to use the fact that every finite subcollection of C has a nonempty intersection, for example.

I want to use this as a lemma to prove the real problem, which is this: "Let [tex]{F_n}[/tex] be a sequence of nonempty closed sets of real numbers with [tex]F_{n+1} \subset F_n[/tex]. Show that if one of the sets is bounded, then [tex] \bigcap_{i=1}^\infty {F_i} \neq \emptyset [/tex]"

(the instructor said we can use other problems not assigned, but we have to solve those too. if anyone can prove it directly, without using the previous prob, feel free to help out anyway )

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# Homework Help: Problem about sets

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