# Definition:Exhausting Sequence of Sets

## Definition

Let $S$ be a set.

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

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

$(1):\quad \forall k \in \N: S_k \subseteq S_{k + 1}$
$(2):\quad \displaystyle \bigcup_{k \mathop \in \N} S_k = S$

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

That is, it is an increasing sequence of subsets of $S$, whose union is $S$.

It is common to write $\left\langle{S_k}\right\rangle_{k \in \N} \uparrow S$ to indicate an exhausting sequence of sets.

Here, the $\uparrow$ denotes a limit of an increasing sequence.