# sequence of sets convergence

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

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

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

In both cases the less accurate notation

 $A=\lim_{n\longrightarrow\infty}A_{n}$

is also used.

Title sequence of sets convergence SequenceOfSetsConvergence 2013-03-22 16:15:12 2013-03-22 16:15:12 Andrea Ambrosio (7332) Andrea Ambrosio (7332) 8 Andrea Ambrosio (7332) Definition msc 28A05