# Definition:Strictly Increasing/Sequence/Real Sequence

< Definition:Strictly Increasing | Sequence(Redirected from Definition:Strictly Increasing Real Sequence)

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## 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.

## Also see

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

## 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