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]] | + | * [[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.
Contents
[hide]Definition
Formally a sequence (Ai)∞i=1 is a function[1][2], f:N→S 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
- 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)
References
- Jump up ↑ p46 - Introduction To Set Theory, third edition, Jech and Hrbacek
- ↑ Jump up to: 2.0 2.1 p11 - Analysis - Part 1: Elements - Krzysztof Maurin