convergent sequence

A sequence x0,x1,x2, in a metric space (X,d) is a convergent sequence if there exists a point xX such that, for every real number ϵ>0, there exists a natural numberMathworldPlanetmath 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