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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.
Table of Contents
Proofs: 1
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Proofs: 1
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References
Bibliography
- Forster Otto: "Analysis 2, Differentialrechnung im \(\mathbb R^n\), Gewöhnliche Differentialgleichungen", Vieweg Studium, 1984
