Definition:Increasing/Sequence

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

Then a sequence $\left \langle {a_k} \right \rangle_{k \in A}$ of terms of $S$ is increasing iff:


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

Real Sequences
The above definition for sequences is usually applied to real number sequences.

Let $\left \langle {x_n} \right \rangle$ be a sequence in $\R$.

Then $\left \langle {x_n} \right \rangle$ is increasing if


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

Also see

 * Strictly Increasing Sequence
 * Decreasing Sequence
 * Monotone Sequence