Definition:Increasing Sequence of Extended Real-Valued Functions

Definition
Let $S$ be a set, and let $\left({f_n}\right)_{n \in \N}, f_n: S \to \overline{\R}$ be a sequence of extended real-valued functions.

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


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

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

Also see

 * Increasing Sequence of Real-Valued Functions, a similar concept for real-valued functions
 * Increasing Sequence of Mappings, an abstraction where $\overline{\R}$ is replaced by an ordered set $T$