\begin{definition}[Definition:Increasing Sequence of Extended Real-Valued Functions]
Let $S$ be a set.
Let $\sequence {f_n}_{n \mathop \in \N}, f_n: S \to \overline{\R}$ be a sequence of extended real-valued functions.
Then $\sequence {f_n}_{n \mathop \in \N}$ is said to be an '''increasing sequence (of extended real-valued functions)''' {{iff}}:
:$\forall s \in S: \forall m, n \in \N: m \le n \implies \map {f_m} s \le \map {f_n} s$
That is, {{iff}} $m \le n \implies f_m \le f_n$, where the second $\le$ denotes pointwise inequality.
\end{definition}