Difference between revisions of "Sequence"

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* [[Bolzano-Weierstrass theorem]]
 
* [[Bolzano-Weierstrass theorem]]
 
* [[Cauchy sequence]] (Alternatively: [[Cauchy criterion for convergence]])
 
* [[Cauchy sequence]] (Alternatively: [[Cauchy criterion for convergence]])
* [[Convergence of a sequence]]
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* [[Convergence of a sequence]] (Or [[Limit (sequence)]] - the page ''Convergence of a sequence'' is being refactored into it)
  
 
==References==
 
==References==
 
<references/>
 
<references/>
 
{{Definition|Set Theory|Real Analysis|Functional Analysis}}
 
{{Definition|Set Theory|Real Analysis|Functional Analysis}}

Revision as of 15:30, 24 November 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

Notation

To specify that the points of a sequence, the x_i are from a space, X we may write:

  • (x_n)^\infty_{n=1}\subseteq X

This is an abuse of notation, as (x_n)^\infty_{n=1} is not a subset of X. It plays on:

  • [(x_n)^\infty_{n=1}\subseteq X]\iff[x\in(x_n)_{n=1}^\infty\implies x\in X]

Note that the elements of (x_n)_{n=1}^\infty are ether:

  • Elements of a relation (if we consider the sequence as a mapping) or
    • So using this, x\in(x_n)_{n=1}^\infty may look like x=(a,b) (indicating f(a)=b) which is an Ordered pair, not in X
  • Elements of a tuple (which is a generalisation of ordered pairs where (usually) (a,b)=\{\{a\},\{a,b\}\}
    • So using this, x\in(x_n)_{n=1}^\infty may indeed look like x=\{\{a\},\{a,b\}\}\notin X

As such the notation (x_n)^\infty_{n=1}\subseteq X having no other sensible meaning is a notation to say that \forall i[x_i\in X]

Subsequence

Given a sequence (x_n)_{n=1}^\infty we define a subsequence of (x_n)^\infty_{n=1}[2] as a sequence:

  • k:\mathbb{N}\rightarrow\mathbb{N} which operates on an n\in\mathbb{N} with n\mapsto k_n:=k(n) where:
    • k_n is increasing, that means k_n\le k_{n+1}

We denote this:

  • (x_{k_n})_{n=1}^\infty

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