Definition:Increasing Sequence of Sets

Definition
Let $X$ be a set, and let $\mathcal S \subseteq \mathcal P \left({X}\right)$ be a collection of subsets of $X$.

An increasing sequence of sets (in $\mathcal S$) is a sequence $\left({A_n}\right)_{n\in \N}$ in $\mathcal S$, such that:


 * $\forall n \in \N: A_n \subseteq A_{n+1}$