# Definition:Decreasing/Sequence

< Definition:Decreasing(Redirected from Definition:Decreasing Sequence)

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## 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 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

## Sources

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