The convergence of a sequence of sets A sequence $\{A_n : n=0,1,2,...\}$ is said to be monotone nondecreasing if we have
$$A_0\subseteq A_1\subseteq \cdot \cdot \cdot \subseteq A_n \subseteq \cdot \cdot \cdot $$
The same sequence is said to be monotone nonincreasing if we have 
$$A_0\supseteq A_1\supseteq \cdot \cdot \cdot \supseteq A_n \supseteq \cdot \cdot \cdot$$
To specify the type of convergence, for a monotone nondecreasing sequence of set we write we write $\{A_{n}\}\uparrow A$, likewise for a monotone nonincreasing sequence we write $\{A_{n}\}\downarrow A$. 
I am confused by the notation, does the latter statement about convergence imply that $\bigcup \{A_n\}=A$? Or, does it only mean that the as $n\to\infty$ the $n$-th element of $\{A_n\}$ will get closer and closer to the some set $A$ (not necessarily $\bigcup\{A_n\}$) but it will never be equal to $A$ because the sequence is infinite? 
 A: The down- and up-arrows imply the monotonicity of the sequence; the $A$ is the actual limit, which is the intersection for a non-increasing sequence and the union for a non-decreasing sequence.
If $\langle A_n:n\in\Bbb N\rangle$ is monotone non-increasing, then its limit is $$A=\bigcap_{n\in\Bbb N}A_n\;,$$ and we write $$\langle A_n:n\in\Bbb N\rangle\downarrow A\;.$$ If the sequence is monotone non-decreasing, then its limit is $$A=\bigcup_{n\in\Bbb N}A_n\;,$$ and we write $$\langle A_n:n\in\Bbb N\rangle\uparrow A\;.$$
A: I think it would be stated somewhere that the union of all $A_n$ is $A$ in the monotone nondecreasing case, and that in the monotone nonincreasing case, the intersection of all $A_n$ is $A$. It would have to be defined this way, not proved this way, so you should look to your book to make sure this is the definition given!
If it is defined this way, as it likely is, then in the nondecreasing case $\bigcup A_n=A$ is correct. If you're having trouble understanding this statement, it can be transformed into a set theoretic one: $x\in A$ iff $\exists n$ such that $x\in A_n$. In the nonincreasing case, we have $A=\bigcap A_n$ meaning $x \in A$ iff $\forall n, x\in A_n$.
A: Often the general notion of "limit" of a sequence of sets $\{A_n : n \in \mathbb{N}\}$ is defined by saying


*

*$x \in \lim_n A_n$ if $x \in A_n$ for all sufficiently large $n$

*$x \notin \lim_n A_n$ if $x \notin A_n$ for all sufficiently large $n$.
If there is some $x$ such that the truth value of "$x \in A_n$" is not eventually constant, then the limit is not defined.
This is the appropriate definition because it makes the characteristic function of $\lim_n A_n$ equal to the pointwise limit of the characteristic functions of the $A_n$'s (and undefined when this pointwise limit is undefined.)
One can show that if the sequence is monotone then the limit exists and is equal to the union (if the sequence is increasing) or the intersection (if the sequence is decreasing.)  These are special cases of the general definition.
