Sequence

A sequence in a set $X$ is a function $a:\mathbb{N}\to X$, where $\mathbb{N}$ is the natural numbers . One writes $a_n$ for $a(n)$, so the sequence is denoted $(a_n)_{n\in\mathbb{N}}$.

Thus a sequence is a function with domain $\mathbb{N}$ and codomain $X$, and it can be manipulated using standard function constructions such as restriction and composition .

Examples:

• The rule $a_n = 1/(n+1)$ defines a sequence of rational numbers .
• The rule $a_n=7$ for all $n\in\mathbb{N}$ defines a constant sequence in $\mathbb{Z}$ (the integers ).