Definition:Increasing Sequence of Mappings

Definition
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) :


 * $\forall s \in S: \forall m, n \in \N: m \le n \implies \map {f_m} s \preceq \map {f_n} s$

That is, $m \le n \implies f_m \preceq f_n$, 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