Sequence

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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.

Definition

Formally a sequence (Ai)i=1 is a function[1][2], f:NS

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:NN which operates on an nN with nkn:=k(n) where:
    • kn is increasing, that means knkn+1

We denote this:

  • (xkn)n=1

See also

References

  1. Jump up p46 - Introduction To Set Theory, third edition, Jech and Hrbacek
  2. Jump up to: 2.0 2.1 p11 - Analysis - Part 1: Elements - Krzysztof Maurin