Definition:Monotone Sequence of Sets

Definition
Let $X$ be a set, and let $\mathcal S \subseteq \mathcal P \left({X}\right)$ be a collection of subsets of $X$.

A monotone sequence of sets (in $\mathcal S$) is a sequence $\left({A_n}\right)_{n \in \N}$ in $\mathcal S$, such that either:
 * $\forall n \in \N: A_n \subseteq A_{n+1}$

or:
 * $\forall n \in \N: A_n \subseteq A_{n+1}$

That is, such that $\left({A_n}\right)_{n \in \N}$ is either increasing or decreasing

Also see

 * Definition:Exhausting Sequence of Sets