Sequences

Why

We want to speak of infinite processes, and to do so we define sequences indexed by $\N $. In other words, important families are those indexed by the natural numbers.

Definition

A sequence (or infinite sequence) is a family whose index set is $\N $ (the set of natural numbers without zero). The $n$th term or coordinate of a sequence is the result of the $n$th natural number, $n \in \N $.¹

Notation

Let $A$ be a non-empty set and $a: \N \to A$. Then $a$ is a (infinite) sequence in $A$. $a(n)$ is the $n$th term. We also denote $a$ by $\seq{a}$ and $a(n)$ by $\seqt{a}$. If $\set{A_n}_{n \in \N }$ is an infinite sequence of sets, then we denote the direct product of the sequence by $\prod_{i = 1}^{\infty} A_i$.

Sometimes the set of infinite sequences in $A$ are denoted $A^\N $ or $A^\infty$.

Natural unions and intersections

We denote the family of the infinite sequence of sets $\seq{A}$ by $\cup_{i = 1}^{\infty} A_i$. Similarly, we denote the intersection of an infinite sequence of sets by $\cap _{i = 1}^{\infty} A_i$, respectively.

1. Future editions may also comment that we are introducing language for the steps of an infinite process. ↩︎