# 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