nested sphere theorem
In a metric space , let be the closedball centered at with radius .
Theorem 1 (Nested sphere theorem [KF]).
A metric space is complete if and only if every sequence
such that and when has a nonempty intersection
(i.e. ).
References
- KF Kolmogorov, A.N. & Fomin, S.V.: Introductory Real Analysis, Translated & Edited by Richard A. Silverman. Dover Publications, Inc. New York, 1970.