# Definition:Strictly 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 **strictly decreasing** if and only if:

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

### 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 **strictly decreasing** if and only if:

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

## Also known as

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

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

Some sources refer to a **strictly decreasing sequence** as a **decreasing sequence**, and refer to a decreasing sequence which is *not* **strictly decreasing** 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

- Definition:Decreasing Sequence
- Definition:Strictly Increasing Sequence
- Definition:Strictly Monotone Sequence

## 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**