Definition:Limit of Increasing Sequence of Sets

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

Let $A = \displaystyle \bigcup_{n \in \N} A_n$.

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