Definition:Limit of Increasing Sequence of Sets

Definition
Let $\left({S_n}\right)_{n \in \N}$ be an increasing sequence of sets.

Let $S = \displaystyle \bigcup_{n \mathop \in \N} S_n$.

Then $S$ is said to be the limit of $\left({S_n}\right)_{n \in \N}$, and one writes $S_n \uparrow S$.