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.  
  
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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.
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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==
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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]]
  
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There is little more to say.
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==Notation==
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To specify that the points of a sequence, the {{M|x_i}} are from a space, {{M|X}} we may write:
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* {{M|1=(x_n)^\infty_{n=1}\subseteq X}}
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This is an abuse of notation, as {{M|1=(x_n)^\infty_{n=1} }} is not a subset of {{M|X}}. It plays on:
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* {{M|1=[(x_n)^\infty_{n=1}\subseteq X]\iff[x\in(x_n)_{n=1}^\infty\implies x\in X]}}
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Note that the elements of {{M|1=(x_n)_{n=1}^\infty}} are ether:
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* Elements of a [[Relation|relation]] (if we consider the sequence as a [[Function|mapping]]) or
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** So using this, {{M|1=x\in(x_n)_{n=1}^\infty}} may look like {{M|1=x=(a,b)}} (indicating {{M|1=f(a)=b}}) which is an [[Ordered pair]], not in {{M|X}}
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* Elements of a [[Tuple|tuple]] (which is a generalisation of [[Ordered pair|ordered pairs]] where (usually) {{M|1=(a,b)=\{\{a\},\{a,b\}\} }}
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** So using this, {{M|1=x\in(x_n)_{n=1}^\infty}} may indeed look like {{M|1=x=\{\{a\},\{a,b\}\}\notin X}}
  
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==Convergence of a sequence==
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'''As such the notation {{M|1=(x_n)^\infty_{n=1}\subseteq X}} having no ''other'' sensible meaning''' is a notation to say that {{M|1=\forall i[x_i\in X]}}
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===Topological form===
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==[[Subsequence]]==
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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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{{:Subsequence/Definition}}
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===Metric space form===
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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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<math>\forall\epsilon>0\exists N\in\mathbb{N}:n > N\implies d(a_n,a)<\epsilon</math> - note the [[Implicit qualifier|implicit <math>\forall n</math>]]
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In this case we may write: <math>\lim_{n\rightarrow\infty}(a_n)=a</math>
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===Basic form===
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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.
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===Normed form===
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In a [[Norm|normed]] [[Vector space|vector space]] as you'd expect it's defined as follows:
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<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
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==See also==
 
==See also==
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* [[Cauchy criterion for convergence]]
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* [[Subsequence]]
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* [[Monotonic sequence]]
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* [[Bolzano-Weierstrass theorem]]
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* [[Cauchy sequence]] (Alternatively: [[Cauchy criterion for convergence]])
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* [[Convergence of a sequence]] (Or [[Limit (sequence)]] - the page ''Convergence of a sequence'' is being refactored into it)
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==Notes==
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<references group="Note"/>
 
==References==
 
==References==
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<references/>
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{{Sequences navbox|plain}}
 
{{Definition|Set Theory|Real Analysis|Functional Analysis}}
 
{{Definition|Set Theory|Real Analysis|Functional Analysis}}
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[[Category:First-year friendly]]

Latest revision as of 18:12, 13 March 2016

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

Definition

Formally a sequence [ilmath](A_i)_{i=1}^\infty[/ilmath] is a function[1][2], [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]. Now we can write:

  • [ilmath]f(i):=A_i[/ilmath]

This naturally then generalises to indexing sets

Notation

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

  • [ilmath](x_n)^\infty_{n=1}\subseteq X[/ilmath]

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

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

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

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

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

Subsequence

Given a sequence [ilmath](x_n)_{n=1}^\infty[/ilmath] we define a subsequence of [ilmath](x_n)^\infty_{n=1}[/ilmath][3][4] as follows:

  • Given any strictly increasing monotonic sequence[Note 1], [ilmath](k_n)_{n=1}^\infty\subseteq\mathbb{N}[/ilmath]
    • That means that [ilmath]\forall n\in\mathbb{N}[k_n<k_{n+1}][/ilmath][Note 2]

Then the subsequence of [ilmath](x_n)[/ilmath] given by [ilmath](k_n)[/ilmath] is:

  • [ilmath](x_{k_n})_{n=1}^\infty[/ilmath], the sequence whose terms are: [ilmath]x_{k_1},x_{k_2},\ldots,x_{k_n},\ldots[/ilmath]
    • That is to say the [ilmath]i[/ilmath]th element of [ilmath](x_{k_n})[/ilmath] is the [ilmath]k_i[/ilmath]th element of [ilmath](x_n)[/ilmath]

As a mapping

Consider an (injective) mapping: [ilmath]k:\mathbb{N}\rightarrow\mathbb{N} [/ilmath] with the property that:

  • [ilmath]\forall a,b\in\mathbb{N}[a<b\implies k(a)<k(b)][/ilmath]

This defines a sequence, [ilmath](k_n)_{n=1}^\infty[/ilmath] given by [ilmath]k_n:= k(n)[/ilmath]

  • Now [ilmath](x_{k_n})_{n=1}^\infty[/ilmath] is a subsequence


See also

Notes

  1. ↑ Note that strictly increasing cannot be replaced by non-decreasing as the sequence could stay the same (ie a term where [ilmath]m_i\eq m_{i+1} [/ilmath] for example), it didn't decrease, but it didn't increase either. It must be STRICTLY increasing.

    If it was simply "non-decreasing" or just "increasing" then we could define: [ilmath]k_n:\eq 5[/ilmath] for all [ilmath]n[/ilmath].
    • Then [ilmath](x_{k_n})_{n\in\mathbb{N} } [/ilmath] is a constant sequence where every term is [ilmath]x_5[/ilmath] - the 5th term of [ilmath](x_n)[/ilmath].
  2. ↑ Some books may simply require increasing, this is wrong. Take the theorem from Equivalent statements to compactness of a metric space which states that a metric space is compact [ilmath]\iff[/ilmath] every sequence contains a convergent subequence. If we only require that:
    • [ilmath]k_n\le k_{n+1} [/ilmath]
    Then we can define the sequence: [ilmath]k_n:=1[/ilmath]. This defines the subsequence [ilmath]x_1,x_1,x_1,\ldots x_1,\ldots[/ilmath] of [ilmath](x_n)_{n=1}^\infty[/ilmath] which obviously converges. This defeats the purpose of subsequences. A subsequence should preserve the "forwardness" of a sequence, that is for a sub-sequence the terms are seen in the same order they would be seen in the parent sequence, and also the "sub" part means building a sequence from it, we want to built a sequence by choosing terms, suggesting we ought not use terms twice.
    The mapping definition directly supports this, as the mapping can be thought of as choosing terms

References

  1. ↑ p46 - Introduction To Set Theory, third edition, Jech and Hrbacek
  2. ↑ p11 - Analysis - Part 1: Elements - Krzysztof Maurin
  3. ↑ Analysis - Part 1: Elements - Krzysztof Maurin
  4. ↑ Functional Analysis - Volume 1: A gentle introduction - Dzung Minh Ha

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