Definition:Increasing Sequence of Extended Real-Valued Functions

Definition

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) if and only if:

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

That is, if and only if $m \le n \implies f_m \le f_n$, 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 $T$