By definition $\limsup_{n\rightarrow\infty}A_n = \cap_{n\geq1} \cup_{k\geq n}A_k$
In words it is
The set of all elements that belong to an infinity of $A_n$ at the exception of a finite number of them.
But, according to the definition, the union when $k$ is greater than $n$ and $n$ goes to infinity this union will only contain the set $A_{\infty}$ then the intersection of all such sets would be just the set $A_{\infty}$, what did I get wrong ?
EDIT: I've made a mistake in verbal formulation of $\limsup$ $\liminf$, here the $\limsup$ is the set of elements that belong to an infinity of $A_n$.
I can't really understand your definition of $A_\infty$, but I can explain what the lim sup is. We have $$\limsup_{n\to\infty}A_n=\bigcap_{n\ge1}\bigcup_{k\ge n}A_k$$
By definition, the elements of the lim sup are those elements in every one of the unions on the right-hand side. That means that if $a$ is an element of the lim sup, then no matter how big $n$ is, there is some $k\ge n$ such that $a\in A_k$. That is, there is a sequence $n_k$ tending to $\infty$ such that $a\in A_{n_k}$ for every $k$.
That is to say, the lim sup is the set of all elements that belong to infinitely many of the $A_k$.
The set of elements that belong to all but finitely many of the $A_k$ is the $$\liminf_{n\to\infty}A_n=\bigcup_{n\ge1}\bigcap_{k\ge n}A_k$$
The right-hand side tells us that an element of the lim inf belongs to all the $A_K$ with $K\ge n$ for some $n$. That is, it belongs to all the $A_k$ except perhaps $A_1,A_2,\dots,A_{n-1}.$