# Definition:Decreasing/Sequence

## Definition

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

Then a sequence $\sequence {a_k}_{k \mathop \in A}$ of terms of $S$ is **decreasing** if and only if:

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

### Real Sequence

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

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

Then $\sequence {x_n}$ is **decreasing** if and only if:

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

## Also known as

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

Some sources use the term **descending sequence**.

Some sources refer to a **decreasing sequence** which is not **strictly decreasing** as **non-increasing** or **monotone non-increasing**.

Some sources refer to a **decreasing sequence** as a **monotonic decreasing sequence** to distinguish it from a **strictly decreasing sequence**.

That is, such that **monotonic** is being used to mean a **decreasing sequence** in which consecutive terms may be equal.

$\mathsf{Pr} \infty \mathsf{fWiki}$ does not endorse this viewpoint.

## Also see

- Results about
**decreasing sequences**can be found**here**.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**decreasing sequence** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**decreasing sequence**