nested sphere theorem

In a metric space $X$, let ${\overline{B}}_r(x)$ be the closed ball centered at $x\in X$ with radius $r>0$.

Theorem 1 (Nested sphere theorem [KF]).

A metric space $X$ is complete if and only if every sequence ${\mathrm{\{}{\overline{B}}_{r_n}\mathit{}\mathrm{(}{x}_n\mathrm{)}\mathrm{\}}}_n$ such that ${\overline{B}}_{r_{i\mathrm{+}\mathrm{1}}}\mathit{}\mathrm{(}{x}_{i\mathrm{+}\mathrm{1}}\mathrm{)}\mathrm{\subseteq }{\overline{B}}_{r_i}\mathit{}\mathrm{(}{x}_i\mathrm{)}$ and $r_n\mathrm{\to }\mathrm{0}$ when $n\mathrm{\to }\mathrm{\infty }$ has a nonempty intersection (i.e. ${\mathrm{\bigcap }}_{n\mathrm{=}\mathrm{1}}^{\mathrm{\infty }}{\overline{B}}_{r_n}\mathit{}\mathrm{(}{x}_n\mathrm{)}\mathrm{\ne }\mathrm{\varnothing }$).

Title	nested sphere theorem
Canonical name	NestedSphereTheorem
Date of creation	2013-03-22 14:57:12
Last modified on	2013-03-22 14:57:12
Owner	Daume (40)
Last modified by	Daume (40)
Numerical id	5
Author	Daume (40)
Entry type	Theorem
Classification	msc 54E35
Classification	msc 54E50