# Definition:Strictly Increasing/Sequence/Real Sequence

## Definition

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

Then $\sequence {x_n}$ is strictly increasing if and only if:

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

## Also known as

A strictly increasing sequence is also referred to as ascending or strictly ascending.

Some sources refer to a strictly increasing sequence as an increasing sequence, and refer to an increasing sequence which is not strictly increasing as a monotonic increasing sequence to distinguish it from a strictly increasing sequence.

That is, such that monotonic is being used to mean an increasing sequence in which consecutive terms may be equal.

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

## Examples

### Example: $\sequence {2^n}$

The first few terms of the real sequence:

$S = \sequence {2^n}_{n \mathop \ge 1}$

are:

$2, 4, 8, 16, \dotsc$

$S$ is strictly increasing.