Difference between revisions of "Sequence"

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==Introduction==
 
 
A sequence is one of the earliest and easiest definitions encountered, but I will restate it.  
 
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.
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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. This notation is inline with that of a [[Tuple|tuple]] which is a generalisation of [[Ordered pair|an ordered pair]].
  
 
==Definition==
 
==Definition==
Formally a sequence is a function<ref>p46 - Introduction To Set Theory, third edition, Jech and Hrbacek</ref>, <math>f:\mathbb{N}\rightarrow S</math> where {{M|S}} is some set. For a finite sequence it is simply <math>f:\{1,...,n\}\rightarrow S</math>
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Formally a sequence {{M|1=(A_i)_{i=1}^\infty}} is a function<ref>p46 - Introduction To Set Theory, third edition, Jech and Hrbacek</ref><ref name="Analysis">p11 - Analysis - Part 1: Elements - Krzysztof Maurin</ref>, <math>f:\mathbb{N}\rightarrow S</math> where {{M|S}} is some set. For a finite sequence it is simply <math>f:\{1,...,n\}\rightarrow S</math>. Now we can write:
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* {{M|1=f(i):=A_i}}
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This naturally then generalises to [[Indexing set|indexing sets]]
  
There is little more to say.
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==Subsequence==
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Given a sequence {{M|1=(x_n)_{n=1}^\infty}} we define a ''subsequence of {{M|1=(x_n)^\infty_{n=1} }}''<ref name="Analysis"/> as a sequence:
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* {{M|k:\mathbb{N}\rightarrow\mathbb{N} }} which operates on an {{M|n\in\mathbb{N} }} with {{M|1=n\mapsto k_n:=k(n)}} where:
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** {{M|k_n}} is increasing, that means {{M|k_n\le k_{n+1} }}
  
==Convergence of a sequence==
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We denote this:
* See [[Convergence of a sequence]]
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* {{M|1=(x_{k_n})_{n=1}^\infty}}
  
 
==See also==
 
==See also==
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* [[Monotonic sequence]]
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* [[Bolzano-Weierstrass theorem]]
 
* [[Cauchy criterion for convergence]]
 
* [[Cauchy criterion for convergence]]
 
* [[Convergence of a sequence]]
 
* [[Convergence of a sequence]]
  
 
==References==
 
==References==
 
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<references/>
 
{{Definition|Set Theory|Real Analysis|Functional Analysis}}
 
{{Definition|Set Theory|Real Analysis|Functional Analysis}}

Revision as of 23:47, 21 June 2015

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. This notation is inline with that of a tuple which is a generalisation of an ordered pair.

Definition

Formally a sequence (Ai)i=1 is a function[1][2], f:NS where S is some set. For a finite sequence it is simply f:{1,...,n}S. Now we can write:

  • f(i):=Ai

This naturally then generalises to indexing sets

Subsequence

Given a sequence (xn)n=1 we define a subsequence of (xn)n=1[2] as a sequence:

  • k:NN which operates on an nN with nkn:=k(n) where:
    • kn is increasing, that means knkn+1

We denote this:

  • (xkn)n=1

See also

References

  1. Jump up p46 - Introduction To Set Theory, third edition, Jech and Hrbacek
  2. Jump up to: 2.0 2.1 p11 - Analysis - Part 1: Elements - Krzysztof Maurin