Definition:Exhausting Sequence of Sets

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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.

Also see