# Definition:Increasing/Sequence

## Definition

Let $\struct {S, \preceq}$ be a totally ordered set.

Let $A$ be a subset of the natural numbers $\N$.

Then a sequence $\sequence {a_k}_{k \mathop \in A}$ of terms of $S$ is **increasing** if and only if:

- $\forall j, k \in A: j < k \implies a_j \preceq a_k$

### Real Sequence

The above definition for sequences is usually applied to real number sequences:

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.

## Also see

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

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

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**increasing sequence** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**increasing sequence** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**increasing sequence**