Definition:Nested Sequence

Definition
Let $S$ be a set.

Let $\mathcal S = \mathcal P \left({S}\right)$ be the power set of $S$.

Let $\left\langle{S_k}\right \rangle_{k \in \N}$ indexed family of subsets of $S$ such that either:
 * $\forall k \in \N: S_k \subseteq S_{k + 1}$

or:
 * $\forall k \in \N: S_k \supseteq S_{k + 1}$

Then $\left\langle{S_k}\right \rangle_{k \in \N}$ is a nested sequence of sets.