Sequence

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Introduction

A sequence is one of the earliest and easiest definitions encountered, but I will restate it.

I was taught to denote the sequence {a1,a2,...}

by {an}n=1
however I don't like this, as it looks like a set. I have seen the notation (an)n=1
and I must say I prefer it.

Definition

Formally a sequence is a function[1], f:NS

where S is some set. For a finite sequence it is simply f:{1,...,n}S

There is little more to say.

Convergence of a sequence

A sequence (an)n=1

in a metric space V (Keep in mind it is easy to get a metric given a normed vector space) is said to converge to a limit aV
if:

ϵ>0NN:n>Nd(an,a)<ϵ

- note the implicit n

In this case we may write: limn(an)=a

Notes

Usually ϵ>0NN:n>N|ana|<ϵ

is first seen, or even just a Null sequence then defining converging to a by subtraction, like with Continuous map you move on to a metric space.


In a normed vector space as you'd expect it's defined as follows:

ϵ>0NN:n>Nana<ϵ

, note this it the definition of the sequence (ana)n=1
tending towards 0

See also

References

  1. Jump up p46 - Introduction To Set Theory, third edition, Jech and Hrbacek