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 the set of natural numbers $\N$.

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

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

In such cases, the natural numbers $\N$ are defined (usually) by the von Neumann construction, resulting in the fact that the two definitions are in complete agreement.

Note, however, that this definition of sequence extends to the transfinite ordinals.

Some sources define a sequence as a succession of numbers only, particularly those addressing analysis.

Also known as
Some sources refer to a sequence as a series.

This usage is not endorsed on, as that term is used to mean something different.

Also see

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


 * Definition:Ordinal Sequence, where the domain is an ordinal, not necessarily $\N$ or a subset thereof


 * Definition:Arithmetic Function, which can be considered as an example of an infinite sequence