Definition:Increasing Sequence of Sets

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Let $S$ be a set.

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

Let $\left\langle{S_k}\right \rangle_{k \in \N}$ 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 $\mathcal S$).

Also known as

Some sources refer to such a sequence of sets as monotone increasing.

Also see