I am trying to understand whether the limit infimum of a set is related or a generalization of the limit infimum of a sequence of real numbers. Suppose that $X_n$ is a sequence of sets, and so the limit infimum of $X_n$ is defined as saying that from some stage onwards, all of the $X_n$ occur.
Concretely:
$$ \liminf_{n \to \infty}X_n = \bigcup_{n=1}^{\infty}\bigcap_{k\geq n}X_k $$
Now, suppose that we let $X_n(\omega) = (-1)^n$ and let $X(\omega) = 1$. Then, we have that:
$$ \liminf_{n \to \infty}X_n = -1 $$
Here it appears that after some time, only $-1$ occurs. But, I know that both $1$ and $-1$ alternate, and so there shouldn't be a time after which $-1$ only occurs.
I am wondering if I am somehow confusing something here between the limit of sets and the limit of a sequence? It also seems that the definition of the limit infimum/supremum of sets has no direct connection to that of a sequence and is an arbitrary definition. Am I wrong here?