Theorem: Nested Closed Subset Theorem 

Let \(X\) be a complete metric space, and let $$A_0\supset A_1\supset A_2\supset A_3\supset \ldots$$ be a sequence of non-empty subsets of \(X\) with diameters converging against \(0\), formally
$$\lim_{k\to\infty}\operatorname{diam}(A_k)=0.$$
Then the intersection of all of these subsets a single point.

| | | | | created: 2014-02-20 22:23:51 | modified: 2017-02-26 00:24:52 | by: bookofproofs | references: [582]

1.Proof: (related to "Nested Closed Subset Theorem")