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

Topological form

A sequence (an)n=1 in a topological space X converges if U that are open neighbourhoods of x NN:n>NxnU

Metric space form

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: lim

Basic form

Usually \forall\epsilon>0\exists N\in\mathbb{N}: n > N\implies |a_n-a|<\epsilon 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.

Normed form

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

\forall\epsilon>0\exists N\in\mathbb{N}:n > N\implies\|a_n-a\|<\epsilon, note this it the definition of the sequence (\|a_n-a\|)_{n=1}^\infty tending towards 0

See also

References

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