# 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.