Difference between revisions of "Sequence"

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(Created page with " ==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,...\}</ma...")
 
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==Convergence of a sequence==
 
==Convergence of a sequence==
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===Topological form===
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A sequence <math>(a_n)_{n=1}^\infty</math> in a [[Topological space|topological space]] {{M|X}} converges if <math>\forall U</math> that are open neighbourhoods of {{M|x}} <math>\exists N\in\mathbb{N}: n> N\implies x_n\in U</math>
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===Metric space form===
 
A sequence <math>(a_n)_{n=1}^\infty</math> in a [[Metric space|metric space]] {{M|V}} (Keep in mind it is easy to get a metric given a [[Norm|normed]] [[Vector space|vector space]]) is said to converge to a limit <math>a\in V</math> if:
 
A sequence <math>(a_n)_{n=1}^\infty</math> in a [[Metric space|metric space]] {{M|V}} (Keep in mind it is easy to get a metric given a [[Norm|normed]] [[Vector space|vector space]]) is said to converge to a limit <math>a\in V</math> if:
  
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In this case we may write: <math>\lim_{n\rightarrow\infty}(a_n)=a</math>
 
In this case we may write: <math>\lim_{n\rightarrow\infty}(a_n)=a</math>
 
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===Basic form===
===Notes===
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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 {{M|a}} by subtraction, like with [[Continuous map]] you move on to a metric space.  
 
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 {{M|a}} by subtraction, like with [[Continuous map]] you move on to a metric space.  
 
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===Normed form===
  
 
In a [[Norm|normed]] [[Vector space|vector space]] as you'd expect it's defined as follows:
 
In a [[Norm|normed]] [[Vector space|vector space]] as you'd expect it's defined as follows:

Revision as of 04:52, 8 March 2015

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