Definition:Increasing Sequence of Real-Valued Functions

Definition
Let $S$ be a set.

Let $\sequence {f_n}_{n \mathop \in \N}, f_n: S \to \R$ be a sequence of real-valued functions.

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


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

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

Also see

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