# Definition:Decreasing/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 decreasing iff:

$\forall j, k \in A: j < k \implies a_k \preceq a_j$

### Real Sequence

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

Let $\left \langle {x_n} \right \rangle$ be a sequence in $\R$.

Then $\left \langle {x_n} \right \rangle$ is decreasing iff:

$\forall n \in \N: x_{n+1} \le x_n$

## Also known as

A decreasing sequence is also referred to as order-reversing.