Let $(a_n) = (1, -1, \frac{1}{2}, 1, -1, \frac{1}{3}, 1, -1, \frac{1}{4}, ...)$
Then $\limsup (a_n) = \inf_n \sup_{k \geq n} (a_n)$
Which in this case is $1.$
However couldn't $(a_n$) be viewed as a sequence of one element sets in which case:
$\limsup (a_n) = \bigcap_n\bigcup_{k=n} a_n$
And this means $x \in a_n$ such that $x$ occurs infinitely often. Which in this case would be: $-1, 1$.
Someone correct my understanding of this and explain the difference between the lim sup of a sequence of numbers vs the lim sup of a sequence of sets.