# reference to Cantor's intersection theorem in complete metric space.

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?

• Rather the diameter tends to 0. – user370967 Sep 3 '18 at 22:04

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.

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