Definition 2

A sequence (resp. a sequence of elements of a set E) is a family (resp. a family of elements of E) whose index set is a subset of $\mathbf{N}$. The sequence is said to be infinite if its index set is an infinite subset of $\mathbf{N}$.

Let $\mathrm{P}\{n\}$ be a relation and let I denote the set of integers $n$ such that $\mathrm{P}\{n\}$ is true. I is then a subset of $\mathbf{N}$. A sequence $(x_n)_{n\in\mathrm{I}}$ is then sometimes written $(x_n)_{\mathrm{P}\{n\}}$, and $x_n$ is called the nth term in the sequence. A sequence whose index set is the set of integers $n \geqslant k$ is often written $(x_n)_{k \leqslant n}$ or $(x_n)_{n \geqslant k}$, or even just $(x_n)$ if $k = 0$ or $k = 1$. Under the same conditions, for example, the notations $\prod_{\mathrm{P}\{n\}} \mathrm{X}_n$ and $\prod_{n=k}^{\infty} \mathrm{X}_n$ are used to denote the product of a sequence of sets $(\mathrm{X}_n)_{n\in\mathrm{I}}$, and there are analogous notations for unions, intersections, cardinal products, and cardinal sums.

Every subfamily of a sequence is a sequence, called a subsequence of the given sequence.

Two sequences $(x_n)_{n\in\mathrm{I}}$, $(y_n)_{n\in\mathrm{I}}$ with the same index set are said to differ only in the order of their terms if there exists a permutation $f$ of the index set I such that $x_{f(n)} = y_n$ for all $n \in \mathrm{I}$.

A multiple sequence is a family whose index set is a subset of a product $\mathbf{N}^p$ ($p$ an integer) ("double sequence" for $p = 2$, "triple sequence" for $p = 3$, and so on).

Let I be a set equipotent to $\mathbf{N}$ and let $f$ be a bijection of $\mathbf{N}$ onto I. For each family $(x_\iota)_{\iota\in\mathrm{I}}$ indexed by the set I, the sequence $n \to x_{f(n)}$ is said to be obtained by arranging the family $(x_\iota)_{\iota\in\mathrm{I}}$ in the order defined by $f$. The sequences which correspond in this way to two distinct bijections of $\mathbf{N}$ onto I differ only in the order of their terms. For a finite family indexed by a set I of $n$ elements we may similarly define a finite sequence with $[1, n]$ or $[0, n-1]$ as index set, by arranging the family in the order defined by a bijection of one or the other of these intervals onto I.