\begin{definition}[Definition:Limit of Increasing Sequence of Sets]
Let $\sequence {S_n}_{n \mathop \in \N}$ be an increasing sequence of sets.
Let $S = \ds \bigcup_{n \mathop \in \N} S_n$.
Then $S$ is said to be the '''limit''' of $\sequence {S_n}_{n \mathop \in \N}$, and one writes $S_n \uparrow S$.
\end{definition}