Definition:Increasing Sequence of Mappings

Definition
Let $S$ be a set, and let $\left({T, \preceq}\right)$ be an ordered set.

Let $\left({f_n}\right)_{n \in \N}, f_n: S \to T$ be a sequence of mappings.

Then $\left({f_n}\right)_{n \in \N}$ is said to be an increasing sequence (of mappings) iff:


 * $\forall s \in S: \forall m,n \in \N: m \le n \implies f_m \left({s}\right) \preceq f_n \left({s}\right)$

That is, iff $m \le n \implies f_m \preceq f_n$, where $\preceq$ denotes pointwise inequality.

Examples

 * Increasing Sequence of Real-Valued Functions, where $T$ is taken to be $\R$ with its usual ordering
 * Increasing Sequence of Extended Real-Valued Functions, where $T$ is taken to be the extended real numbers $\overline{\R}$ with their ordering