5.2. Compact and Perfect Sets

Proposition 5.2.8: Intersection of Nested Compact Sets

Suppose { Aj } is a collection of sets such that each Ak is non-empty, compact, and Aj+1 Aj. Then A = Aj is not empty.


Each Aj is compact, hence closed and bounded. Therefore, A is closed and bounded as well, and hence A is compact. Pick an aj Aj for each j.

Then the sequence { aj } is contained in A1. Since that set is compact, there exists a convergent subsequence { ajk } with limit in A1.

But that subsequence, except the first number, is also contained in A2. Since A2 is compact, the limit must be contained in A2.

Continuing in this fashion, we see that the limit must be contained in every Aj, and hence it is also contained in their intersection A. But then A can not be empty.

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