# Definition:Nested Sequence

## Definition

Let $S$ be a set.

Let $\mathcal S = \powerset S$ be the power set of $S$.

Let $\family {S_k}_{k \mathop \in \N}$ 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 $\family {S_k}_{k \mathop \in \N}$ is a nested sequence of sets.

### Increasing Sequence

Let $\left\langle{S_k}\right \rangle_{k \in \N}$ be a nested sequence of subsets of $S$ such that:

$\forall k \in \N: S_k \subseteq S_{k + 1}$

Then $\left\langle{S_k}\right \rangle_{k \in \N}$ is an increasing sequence of sets (in $\mathcal S$).

### Decreasing Sequence

Let $\left\langle{S_k}\right \rangle_{k \in \N}$ be a nested sequence of subsets of $S$ such that:

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

Then $\left\langle{S_k}\right \rangle_{k \in \N}$ is a decreasing sequence of sets (in $\mathcal S$).

## Also known as

A nested sequence is also seen referred to as a chain, but as that term is also used for another concept, it is preferred on $\mathsf{Pr} \infty \mathsf{fWiki}$ that the latter is not used.