# 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