# Definition:Monotone (Order Theory)/Sequence

## Definition

Let $\struct {S, \preceq}$ be a totally ordered set.

A sequence $\sequence {a_k}_{k \mathop \in A}$ of elements of $S$ is monotone if and only if it is either increasing or decreasing.

### Real Sequence

The above definition for sequences is usually applied to real number sequences:

Let $\sequence {x_n}$ be a sequence in $\R$.

Then $\sequence {x_n}$ is monotone if and only if it is either increasing or decreasing.

## Also known as

This can also be called a monotonic sequence.