# Definition:Monotone (Order Theory)/Sequence

## Definition

Let $\left({S, \preceq}\right)$ be a totally ordered set.

Then a sequence $\left \langle {a_k} \right \rangle_{k \in A}$ of terms of $S$ is **monotone** 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**.