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?
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:
(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! :)