# Definition:Nested Sequence

## Definition

Let $S$ be a set.

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

Let $\sequence {S_k}_{k \mathop \in \N}$ be a sequence 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 $\sequence {S_k}_{k \mathop \in \N}$ be a nested sequence of subsets of $S$ such that:

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

Then $\sequence {S_k}_{k \mathop \in \N}$ is an increasing sequence of sets (in $\SS$).

### Decreasing Sequence

Let $\sequence {S_k}_{k \mathop \in \N}$ be a nested sequence of subsets of $S$ such that:

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

Then $\sequence {S_k}_{k \mathop \in \N}$ is a decreasing sequence of sets (in $\SS$).

## Also known as

A nested sequence is a specific example of a chain (of sets) in which the underlying set forms a sequence.

Hence it is also a specific example of a nest.

## Also see

• Results about nested sequences can be found here.