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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,...} by \{a_n\}_{n=1}^\infty however I don't like this, as it looks like a set. I have seen the notation (a_n)_{n=1}^\infty 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 (A_i)_{i=1}^\infty is a function[1][2], f:\mathbb{N}\rightarrow S where S is some set. For a finite sequence it is simply f:\{1,...,n\}\rightarrow S. Now we can write:
- f(i):=A_i
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}[3][4] as follows:
- Given any strictly increasing monotonic sequence[Note 1], (k_n)_{n=1}^\infty\subseteq\mathbb{N}
- That means that \forall n\in\mathbb{N}[k_n<k_{n+1}][Note 2]
Then the subsequence of (x_n) given by (k_n) is:
- (x_{k_n})_{n=1}^\infty, the sequence whose terms are: x_{k_1},x_{k_2},\ldots,x_{k_n},\ldots
- That is to say the ith element of (x_{k_n}) is the k_ith element of (x_n)
As a mapping
Consider an (injective) mapping: k:\mathbb{N}\rightarrow\mathbb{N} with the property that:
- \forall a,b\in\mathbb{N}[a<b\implies k(a)<k(b)]
This defines a sequence, (k_n)_{n=1}^\infty given by k_n:= k(n)
- Now (x_{k_n})_{n=1}^\infty 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 m_i\eq m_{i+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: k_n:\eq 5 for all n.- Then (x_{k_n})_{n\in\mathbb{N} } is a constant sequence where every term is x_5 - the 5th term of (x_n).
- 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 \iff every sequence contains a convergent subequence. If we only require that:
- k_n\le k_{n+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