Solving Problem about Sets with Heine-Borel & De Morgan's Law

[fourier jr]

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, \bigcap_{F \in C}^F \neq \emptyset"

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 {F_n} be a sequence of nonempty closed sets of real numbers with F_{n+1} \subset F_n. Show that if one of the sets is bounded, then \bigcap_{i=1}^\infty {F_i} \neq \emptyset"

(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 )

 

[Hurkyl]

Hrm, I would think a proof by contradiction would be the best approach. Assume that the intersection is empty, and try to use compactness to construct a finite subcollection with empty intersection.

 

[shmoe]

For your full problem, you can assume that all the F_n are bounded, they are all bounded after some point and tossing out the (finite number of) earlier unbounded ones will have no effect on your final set.

You know \bigcap_{i=1}^k {F_i} is non-empty for all k. You can use this to build a sequence in F_1. What does it converge to? Can you show this point is in all your sets?