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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==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 {a1,a2,...}
Contents
[hide]Definition
Formally a sequence (Ai)∞i=1 is a function[1][2], f:N→S
- f(i):=Ai
This naturally then generalises to indexing sets
Notation
To specify that the points of a sequence, the xi are from a space, X we may write:
- (xn)∞n=1⊆X
This is an abuse of notation, as (xn)∞n=1 is not a subset of X. It plays on:
- [(xn)∞n=1⊆X]⟺[x∈(xn)∞n=1⟹x∈X]
Note that the elements of (xn)∞n=1 are ether:
- Elements of a relation (if we consider the sequence as a mapping) or
- So using this, x∈(xn)∞n=1 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∈(xn)∞n=1 may indeed look like x={{a},{a,b}}∉X
As such the notation (xn)∞n=1⊆X having no other sensible meaning is a notation to say that ∀i[xi∈X]
Subsequence
Given a sequence (xn)∞n=1 we define a subsequence of (xn)∞n=1[3][4] as follows:
- Given any strictly increasing monotonic sequence[Note 1], (kn)∞n=1⊆N
- That means that ∀n∈N[kn<kn+1][Note 2]
Then the subsequence of (xn) given by (kn) is:
- (xkn)∞n=1, the sequence whose terms are: xk1,xk2,…,xkn,…
- That is to say the ith element of (xkn) is the kith element of (xn)
As a mapping
Consider an (injective) mapping: k:N→N with the property that:
- ∀a,b∈N[a<b⟹k(a)<k(b)]
This defines a sequence, (kn)∞n=1 given by kn:=k(n)
- Now (xkn)∞n=1 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
- Jump up ↑ Note that strictly increasing cannot be replaced by non-decreasing as the sequence could stay the same (ie a term where mi=mi+1 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: kn:=5 for all n.- Then (xkn)n∈N is a constant sequence where every term is x5 - the 5th term of (xn).
- Jump up ↑ 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 ⟺ every sequence contains a convergent subequence. If we only require that:
- kn≤kn+1
The mapping definition directly supports this, as the mapping can be thought of as choosing terms
References
- Jump up ↑ p46 - Introduction To Set Theory, third edition, Jech and Hrbacek
- Jump up ↑ p11 - Analysis - Part 1: Elements - Krzysztof Maurin
- Jump up ↑ Analysis - Part 1: Elements - Krzysztof Maurin
- Jump up ↑ Functional Analysis - Volume 1: A gentle introduction - Dzung Minh Ha