Tough proving question

[trap]

Any help will be very gracious.

If \text C_{1} , C_{2} , C_{3} are all non empty compact sets in \text R^n such that \text C_{k+1} \subset C_{k} for all k=1,2,3,......., then the set \text C = I_{k=1}^{\infty}C_{k} is also non-empty.

[CarlB]

The basic problem is that there are an infinite number of C_n involved. You should already know how to deal with unions of infinite numbers of open sets. So try looking at the complements of C_n.

Carl

[HallsofIvy]

The complement of a closed set is open and of course all compact sets are closed. But it is crucial that these sets be compact and don't see how looking at their complements will include that.

If you were allowed to use the "Finite Intersection Propery"- If the intersection of any finite subset of a collection of compact sets is non-empty, then the intersection of all of them is non-empty- this would be trivial. However, I suspect that the whole point of this is to prove a relatively simple version of that.

Try this: Let {Uα} be an open cover for C1. Then, since every Cn is a subset of C1, it is also an open cover for Cn. Since each Cn is a compact, there exist a finite sub-cover.
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