# Definition:Increasing/Sequence

## Definition

Let $\left({S, \preceq}\right)$ be a totally ordered set.

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

Then a sequence $\left \langle {a_k} \right \rangle_{k \in A}$ 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.