\begin{definition}[Definition:Exhausting Sequence of Sets]
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 nested sequence of subsets of $S$ such that:
:$(1): \quad \forall k \in \N: S_k \subseteq S_{k + 1}$
:$(2): \quad \ds \bigcup_{k \mathop \in \N} S_k = S$
Then $\sequence {S_k}_{k \mathop \in \N}$ is an '''exhausting sequence of sets (in $\SS$)'''.
That is, it is an increasing sequence of subsets of $S$, whose union is $S$.
It is common to write $\sequence {S_k}_{k \mathop \in \N} \uparrow S$ to indicate an '''exhausting sequence of sets'''.
Here, the $\uparrow$ denotes a limit of an increasing sequence.
\end{definition}