Cauchy criterion for convergence
A series in a Banach space is http://planetmath.org/node/2311convergent iff for every there is a number such that
holds for all and .
Proof:
First define
Now, since is complete, converges if and only if it is a Cauchy sequence, so if for every there is a number , such that for all holds:
We can assume and thus set . The series is iff
Title | Cauchy criterion for convergence |
---|---|
Canonical name | CauchyCriterionForConvergence |
Date of creation | 2013-03-22 13:22:03 |
Last modified on | 2013-03-22 13:22:03 |
Owner | mathwizard (128) |
Last modified by | mathwizard (128) |
Numerical id | 14 |
Author | mathwizard (128) |
Entry type | Theorem |
Classification | msc 40A05 |