Difference between revisions of "Sequence"
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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. | + | 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> | + | 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: |
+ | * {{M|1=f(i):=A_i}} | ||
+ | This naturally then generalises to [[Indexing set|indexing sets]] | ||
− | + | ==Notation== | |
+ | To specify that the points of a sequence, the {{M|x_i}} are from a space, {{M|X}} we may write: | ||
+ | * {{M|1=(x_n)^\infty_{n=1}\subseteq X}} | ||
+ | This is an abuse of notation, as {{M|1=(x_n)^\infty_{n=1} }} is not a subset of {{M|X}}. It plays on: | ||
+ | * {{M|1=[(x_n)^\infty_{n=1}\subseteq X]\iff[x\in(x_n)_{n=1}^\infty\implies x\in X]}} | ||
+ | Note that the elements of {{M|1=(x_n)_{n=1}^\infty}} are ether: | ||
+ | * Elements of a [[Relation|relation]] (if we consider the sequence as a [[Function|mapping]]) or | ||
+ | ** 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}} | ||
+ | * Elements of a [[Tuple|tuple]] (which is a generalisation of [[Ordered pair|ordered pairs]] where (usually) {{M|1=(a,b)=\{\{a\},\{a,b\}\} }} | ||
+ | ** So using this, {{M|1=x\in(x_n)_{n=1}^\infty}} may indeed look like {{M|1=x=\{\{a\},\{a,b\}\}\notin X}} | ||
− | + | '''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]}} | |
− | + | ==[[Subsequence]]== | |
− | + | {{:Subsequence/Definition}} | |
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− | == | + | |
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==See also== | ==See also== | ||
− | * [[Cauchy criterion for convergence]] | + | * [[Subsequence]] |
− | + | * [[Monotonic sequence]] | |
+ | * [[Bolzano-Weierstrass theorem]] | ||
+ | * [[Cauchy sequence]] (Alternatively: [[Cauchy criterion for convergence]]) | ||
+ | * [[Convergence of a sequence]] (Or [[Limit (sequence)]] - the page ''Convergence of a sequence'' is being refactored into it) | ||
+ | ==Notes== | ||
+ | <references group="Note"/> | ||
==References== | ==References== | ||
− | + | <references/> | |
+ | {{Sequences navbox|plain}} | ||
{{Definition|Set Theory|Real Analysis|Functional Analysis}} | {{Definition|Set Theory|Real Analysis|Functional Analysis}} | ||
+ | [[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
- Subsequence
- Monotonic sequence
- Bolzano-Weierstrass theorem
- Cauchy sequence (Alternatively: Cauchy criterion for convergence)
- Convergence of a sequence (Or Limit (sequence) - the page Convergence of a sequence is being refactored into it)
Notes
- ↑ 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].
- ↑ 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]
The mapping definition directly supports this, as the mapping can be thought of as choosing terms
References
- ↑ p46 - Introduction To Set Theory, third edition, Jech and Hrbacek
- ↑ p11 - Analysis - Part 1: Elements - Krzysztof Maurin
- ↑ Analysis - Part 1: Elements - Krzysztof Maurin
- ↑ Functional Analysis - Volume 1: A gentle introduction - Dzung Minh Ha