I am referring to the post: lim sup and lim inf of sequence of sets.
how would we prove that for $\limsup A_n $ $x$ is in infinitely many sets $A_i$?
If I define set $A$ as a set that consists of infinitely many sets $x \in $ $A_i$, what is its complement $A^C$ ?
Namely, I would like to prove that
$$ A^c \subseteq \bigcup_{N=1}^\infty \bigcap_{n\ge N} A_n^C $$
In the answer to the post you mention it is explained very nicely that $$x\in\limsup A_n= \bigcap_{N=1}^\infty \bigcup_{n\ge N} A_n\iff x\text{ is contained in }A_n\text{ for infinitely many }n\in\mathbb N$$
So I don't really understand why you are asking this question here.
$x\in(\limsup A_n)^c$ then tells us that $x$ is contained in $A_n$ for a finite number of $n\in\mathbb N$ or equivalently that some $N$ exists with $n\ge N\Rightarrow x\in A_n^c$
That means $x\in\liminf A_n^c=\bigcup_{N=1}^\infty \bigcap_{n\ge N} A_n^c$ wich is also nicely explained in the answer.