Difference between revisions of "Sequence"
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This naturally then generalises to [[Indexing set|indexing sets]] | 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== | ||
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: | 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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* [[Monotonic sequence]] | * [[Monotonic sequence]] | ||
* [[Bolzano-Weierstrass theorem]] | * [[Bolzano-Weierstrass theorem]] | ||
| − | * [[Cauchy criterion for convergence]] | + | * [[Cauchy sequence]] (Alternatively: [[Cauchy criterion for convergence]]) |
* [[Convergence of a sequence]] | * [[Convergence of a sequence]] | ||
Revision as of 13:56, 9 July 2015
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][2] as a sequence:
- [ilmath]k:\mathbb{N}\rightarrow\mathbb{N} [/ilmath] which operates on an [ilmath]n\in\mathbb{N} [/ilmath] with [ilmath]n\mapsto k_n:=k(n)[/ilmath] where:
- [ilmath]k_n[/ilmath] is increasing, that means [ilmath]k_n\le k_{n+1} [/ilmath]
We denote this:
- [ilmath](x_{k_n})_{n=1}^\infty[/ilmath]
See also
- Monotonic sequence
- Bolzano-Weierstrass theorem
- Cauchy sequence (Alternatively: Cauchy criterion for convergence)
- Convergence of a sequence
