Is this how the limit of a sequence of sets is commonly defined? I was looking at the wikipedia page for the Cantor set which defined the set using a limit. I had not previously seen a limit expression involving a sequence of sets rather than real numbers, so I got curious as to how this was defined. I tried to think of an adequate definition myself:
$$\lim_{n\to\infty}A_n=B\\\Updownarrow\\\forall ~x\in B~(\exists~K~(\forall~n>K~(x\in A_n)))\\\land\\\forall ~x\notin B~(\exists~K~(\forall~n>K~(x\notin A_n)))$$
Here $x\notin B$ means $x\in X\setminus B$ for some set $X$ which all $A_n$ and $B$ are subsets of. While searching for the commonly used definition, I found this page, but I am not quite sure it is talking about the same thing.

Question: Is this definition equivalent to the commonly used one?

 A: What is definitely allowable is to take limits of nested sets, since then we can write the limit as an infinite union. We can extend this to create a reasonable definition of the limit surperior and limit inferior of sequences of sets. Then if $\limsup A_n=\liminf A_n$, it is sensible to say that $A_n\to A$: this is the notion that your linked page provides.
We can paraphrase your definition as follows:

$x\in B$ if it is in cofinitely many $A_n$, and $x\notin B$ if $x\notin A_n$ for cofinitely many $n$.

(cofinitely means "all but finitely", which is formalized in the standard $\exists K, \forall n> K…$ language.)
Note that $\liminf A_n$ is the set of points which are in cofinitely many $A_n$. So if $x\notin\liminf A$ then $x$ is necessarily only in finitely many $A_n$, or in other words, $x\notin A_n$ for only cofinitely many $A_n$. Moreover, $\limsup A_n$ is the set of points which are in infinitely many $A_n$. Therefore, if $A_n$ has a limit in the usual sense, every point in infinitely many $A_n$ is in fact in cofinitely many. Hence $x\in\lim A_n$ if it is in cofinitely many $A_n$.
Therefore, the definitions are the same.
Good eye! :)
