# Definition:Strictly Decreasing/Sequence/Real Sequence

## Definition

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.

## Examples

### Example: $\sequence {n^{-1} }$

The first few terms of the real sequence:

- $S = \sequence {n^{-1} }_{n \mathop \ge 1}$

are:

- $1, \dfrac 1 2, \dfrac 1 3, \dfrac 1 4, \dotsc$

$S$ is strictly decreasing.

## Also see

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

## Sources

- 1919: Horace Lamb:
*An Elementary Course of Infinitesimal Calculus*(3rd ed.) ... (previous) ... (next): Chapter $\text I$. Continuity: $2$. Upper or Lower Limit of a Sequence - 1971: Robert H. Kasriel:
*Undergraduate Topology*... (previous) ... (next): $\S 1.15$: Sequences - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 4$: Convergent Sequences: $\S 4.15$: Monotone Sequences - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**decreasing sequence**