# 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.