# Definition:Strictly Increasing/Sequence

## Definition

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

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

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

### 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 **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.

## Also see

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

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 18$: Induced $N$-ary Operations - 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**