I'm trying to come up with a sequence of sets $(A_n)_{n\geq1} = A_1, A_2, ... \subset S$, such that for every member $x$ of $\operatorname{lim sup} A_n$ the following two conditions hold:
i) $x \in A_k$ for infinitely many of the $A_k$ (this qualifies $x$ to be a member of $\operatorname{lim sup} A_n$). And also,
ii) $x \notin A_i$ for infinitely many $A_i$
I'm using this definition: $\operatorname{lim sup} A_n = \bigcap_{k=1}^\infty \bigcup_{n=k}^\infty A_n$.
Let the two sets $A,B$ be disjoint and consider the sequence $A,B,A,B,A,B,\cdots$ which alternates. The lim sup is then the union $A \cup B$ while if say $x \in A$ it it in none of the sets $B$ and conversely if $x \in B$ it is in none of the sets $A.$
Note: The sets $A,B$ should be nonempty, or at least one of them.