Sequence
From Maths
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
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 criterion for convergence
- Convergence of a sequence
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