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The Cantor's intersection theorem in the formulation of metric spaces says the following. Assume $A_{n}$ is a sequence of nested and closed subsets in a complete metric space. Assume that $\lim_{n \to \infty} \operatorname{diam}(A_{n})= 0$. Then $\cap_{n\in N} A_{n}$ is a singleton.

I am looking for a reference to the above theorem in some books on topology. I can't find a reference so quickly. May anyone suggest a reference?

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  • $\begingroup$ Rather the diameter tends to 0. $\endgroup$ – user370967 Sep 3 '18 at 22:04
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It's Theorem 4.3.9. in Engelking's General Topology (2nd ed.)

Also Lemma 7.3 (p 295) in Munkres' Topology (first edition), or Lemma 48.3 (p 297) in the second edition.

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"Introduction to Mathematical Analysis" by Steven A. Douglass proves this assuming the archimedean property of the reals.

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