\begin{definition}[Definition:Monotone Sequence of Sets]
Let $X$ be a set.
Let $\SS \subseteq \powerset X$ be a collection of subsets of $X$.
A '''monotone sequence of sets (in $\SS$)''' is a sequence $\sequence {A_n}_{n \mathop \in \N}$ in $\SS$, 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
\end{definition}