\begin{definition}[Definition:Increasing Sequence of Mappings]
Let $S$ be a set, and let $\struct {T, \preceq}$ be an ordered set.
Let $\sequence {f_n}_{n \mathop \in \N}, f_n: S \to T$ be a sequence of mappings.
Then $\sequence {f_n}_{n \mathop \in \N}$ is said to be an '''increasing sequence (of mappings)''' {{iff}}:
:$\forall s \in S: \forall m, n \in \N: m \le n \implies \map {f_m} s \preceq \map {f_n} s$
That is, {{iff}} $m \le n \implies f_m \preceq f_n$, where $\preceq$ denotes pointwise inequality.
\end{definition}