Sequence

Introduction

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

I was taught to denote the sequence [math]\{a_1,a_2,...\}[/math] by [math]\{a_n\}_{n=1}^\infty[/math] however I don't like this, as it looks like a set. I have seen the notation [math](a_n)_{n=1}^\infty[/math] and I must say I prefer it.

Definition

Formally a sequence is a function, [math]f:\mathbb{N}\rightarrow S[/math] where [ilmath]S[/ilmath] is some set. For a finite sequence it is simply [math]f:\{1,...,n\}\rightarrow S[/math]

There is little more to say.

Convergence of a sequence

Topological form

A sequence [math](a_n)_{n=1}^\infty[/math] in a topological space [ilmath]X[/ilmath] converges if [math]\forall U[/math] that are open neighbourhoods of [ilmath]x[/ilmath] [math]\exists N\in\mathbb{N}: n> N\implies x_n\in U[/math]

Metric space form

A sequence [math](a_n)_{n=1}^\infty[/math] in a metric space [ilmath]V[/ilmath] (Keep in mind it is easy to get a metric given a normed vector space) is said to converge to a limit [math]a\in V[/math] if:

[math]\forall\epsilon>0\exists N\in\mathbb{N}:n > N\implies d(a_n,a)<\epsilon[/math] - note the implicit [math]\forall n[/math]

In this case we may write: [math]\lim_{n\rightarrow\infty}(a_n)=a[/math]

Basic form

Usually [math]\forall\epsilon>0\exists N\in\mathbb{N}: n > N\implies |a_n-a|<\epsilon[/math] is first seen, or even just a Null sequence then defining converging to [ilmath]a[/ilmath] 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:

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

See also

• Cauchy criterion for convergence