Definition:Sequence

Informal Definition
A sequence is a set of objects which is listed in a specific order, one after another.

Thus one can identify the elements of a sequence as being the first, the second, the third, ... the $n$th, and so on.

Formal Definition
A sequence is a mapping whose domain is a subset of $\N$.

It can be seen that a sequence is an instance of a family of elements indexed by $\N$.

Codomain
The codomain of a sequence can be elements of a set of any objects.

If the codomain of a sequence $f$ is $S$, then the sequence is said to be a sequence of elements of $S$, or a sequence in $S$.

Notation
The notation for a sequence is as follows.

If $f: A \to S$ is a sequence, then a symbol, for example "$a$", is chosen to represent elements of this sequence.

Then for each $k \in A$, $f \left({k}\right)$ is denoted $a_k$, and $f$ itself is denoted $\left \langle {a_k} \right \rangle_{k \in A}$.

Other types of brackets may be encountered, eg. $\left({a_k}\right)_{k \in A}$ and $\left\{{a_k}\right\}_{k \in A}$.

The latter is discouraged because of the implication that the order of the terms does not matter.

Any expression can be used to denote the domain of $f$ in place of $k \in A$.

The set $A$ is usually understood to be the set $\left\{{1, 2, 3, \ldots, n}\right\}$.

If this is the case, then it is usual to write $\left \langle {a_k} \right \rangle_{k \in A}$ as $\left \langle {a_k} \right \rangle$ or even as $\left \langle {a} \right \rangle$ if brevity and simplicity improve clarity.

Sequence of Distinct Terms
A sequence of distinct terms of $S$ is an injection from a subset of $\N$ into $S$.

A sequence $\left \langle {a_k} \right \rangle_{k \in A}$ is a sequence of distinct terms iff $a_j \ne a_k$ for all $j, k \in A$ such that $j \ne k$.

Equality of Sequences
Let $f$ and $g$ be two sequences:
 * $f = \left({x_1, x_2, \ldots, x_n}\right)$
 * $g = \left({y_1, y_2, \ldots, y_m}\right)$

Then $f = g$ iff:
 * $m = n$
 * $\forall i: 1 \le i \ne n: x_i = y_i$

Also see

 * Rational Sequence
 * Real Sequence
 * Complex Sequence