convergent sequence
A sequence x0,x1,x2,… in a metric space (X,d) is a convergent sequence if there exists a point x∈X such that, for every real number ϵ>0, there exists a natural number N such that d(x,xn)<ϵ for all n>N.
The point x, if it exists, is unique, and is called the limit point or limit of the sequence. One can also say that the sequence x0,x1,x2,… converges to x.
A sequence is said to be divergent if it does not converge.
Title | convergent sequence |
Canonical name | ConvergentSequence |
Date of creation | 2013-03-22 11:55:07 |
Last modified on | 2013-03-22 11:55:07 |
Owner | djao (24) |
Last modified by | djao (24) |
Numerical id | 10 |
Author | djao (24) |
Entry type | Definition |
Classification | msc 54E35 |
Classification | msc 40A05 |
Related topic | AxiomOfAnalysis |
Related topic | BolzanoWeierstrassTheorem |
Related topic | Sequence |
Defines | limit point |
Defines | limit |
Defines | converge |
Defines | diverge |
Defines | divergent sequence |