Definition:Monotone Sequence of Sets

Definition
Let $X$ be a set.

Let $\mathcal S \subseteq \powerset X$ be a collection of subsets of $X$.

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

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

That is, such that $\sequence {A_n}_{n \mathop \in \N}$ is either increasing or decreasing

Also see

 * Definition:Exhausting Sequence of Sets