Convergence of a sequence

From Maths
Revision as of 15:29, 24 November 2015 by Alec (Talk | contribs)

Jump to: navigation, search
This page is currently being refactored (along with many others)
Please note that this does not mean the content is unreliable. It just means the page doesn't conform to the style of the site (usually due to age) or a better way of presenting the information has been discovered.

This page is to be phased out and the content moved to either more appropriate places or to Limit (sequence)

Like with continuity there are three forms for convergence of a Sequence

Given a sequence (an)nn=1

we may say that it converges to a or limn(an)=a
if and only if the following definition holds:

First form

Introductory form ϵ>0NN:n>N|ana|<ϵ

Second form

Metric space form ϵ>0NN:n>Nd(ana)<ϵ

Third form

Topological form NaNN:n>NanNa

where Na
denotes a neighbourhood of a

Cauchy Criterion

Convergence can be shown without knowing what exactly the sequence converges to, see the Cauchy criterion for convergence page

Note on norms

Recall from norm that we can simply define d(x,y)=xy

, thus we can also have a slight variation of the metric form:

ϵ>0NN:n>Nana<ϵ

Is is worth noting because in Functional Analysis norms are considered and if we deal with a metric space we are inside a branch of topology

Interesting examples

fn(t)=tn0
in L1

Using the L1

norm stated here for convenience: fLp=(10|f(x)|pdx)1p
so fL1=10|f(x)|dx

We see that fnL1=10xndx=[1n+1xn+1]10=1n+1

This clearly 0

- this is 0:[0,1]R
which of course has norm 0, we think of this from the sequence (fn0L1)n=10fn0