sequence of sets convergence

Let ${\left\{A_n\right\}}_{n=1}^{\mathrm{\infty }}$ be a sequence of sets,and $A$ a set.

The sequence ${\left\{A_n\right\}}_{n=1}^{\mathrm{\infty }}$ is said to from below to $A$, (shortly, $A_n\uparrow A$ or $A_n\nearrow A$), iff
1) $A_n\subseteq A_{n+1}$  $\forall n\ge 1$
2) $A={\displaystyle \bigcup _{n=1}^{\mathrm{\infty }}}{A}_n$

The sequence ${\left\{A_n\right\}}_{n=1}^{\mathrm{\infty }}$ is said to from above to $A$, (shortly, $A_n\downarrow A$ or $A_n\searrow A$), iff
1) $A_{n+1}\subseteq A_n$  $\forall n\ge 1$
2) $A={\displaystyle \bigcap _{n=1}^{\mathrm{\infty }}}{A}_n$

In both cases the less accurate notation

$$A=\underset{n⟶\mathrm{\infty }}{lim}{A}_n$$

is also used.