\begin{definition}[Definition:Increasing/Sequence]
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''' {{iff}}:
:$\forall j, k \in A: j < k \implies a_j \preceq a_k$
\end{definition}