# Definition:Increasing/Sequence/Real Sequence

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

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## Definition

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

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

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

## Also known as

An **increasing sequence** is also known as an **ascending sequence**.

Some sources refer to an **increasing sequence** which is not **strictly increasing** as **non-decreasing** or **monotone non-decreasing**.

Some sources refer to an **increasing sequence** 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: $\sequence 1$

The first few terms of the real sequence:

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

are:

- $1, 1, 1, 1, \dotsc$

$S$ is both increasing and decreasing.

## Also see

- Definition:Strictly Increasing Real Sequence
- Definition:Decreasing Real Sequence
- Definition: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 - 1962: Bert Mendelson:
*Introduction to Topology*... (previous) ... (next): $\S 2.5$: Limits: Exercise $4$ - 1971: Robert H. Kasriel:
*Undergraduate Topology*... (previous) ... (next): $\S 1.15$: Sequences: Definition $15.3$ - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $\S 1.2$: Real Sequences: Definition $1.2.5$ - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 4$: Convergent Sequences: $\S 4.15$: Monotone Sequences