Sequence
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 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[2] as a sequence:
- k:N→N which operates on an n∈N with n↦kn:=k(n) where:
- kn is increasing, that means kn≤kn+1
We denote this:
- (xkn)∞n=1
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