Let $(A_n)_{n=1}^{\infty}$ be a pairwise disjoint collection.
$\lim A_n = \emptyset$? (see here and there)
What about a set of extended real numbers $A_n=(n,n+1]$?
It seems that
$\limsup A_n = \bigcap_{k=1}^{\infty} \bigcup_{n=k}^{\infty} A_n$
$= \bigcap_{k=1}^{\infty} \bigcup_{n=k}^{\infty} (n, n+1]$
$= \bigcap_{k=1}^{\infty} ((k, k+1] \cup (k+1, k+2] \cup ...)$
$= \bigcap_{k=1}^{\infty} ((k, \infty])$
... = {$\infty$} ?