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 $\N$.

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.

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:
 * $m = n$
 * $\forall i: 1 \le i \le n: x_i = y_i$

Also defined as
Some sources, generally expositions of set theory, define a sequence as a mapping whose domain is an ordinal.

From Ordinal Equivalent to Natural Number the two definitions can be seen to be identical when the ordinals in question are indeed the natural numbers, but note that this definition extends to the transfinite ordinals.

Also see

 * Definition:Rational Sequence
 * Definition:Real Sequence
 * Definition:Complex Sequence