Suppose $n \in \mathbb{N}$ and let $P(n)$ denote a property that holds for all but finitely many $n$. Then clearly $P(n)$ holds infinitely often. More generally, for an infinite sequence of sets $(P_1,P_2,\dots)$ we may conclude that $x \in \lim\inf P_i \implies x \in \lim \sup P_i$, or equivalently:
$$ \bigcup_{i}\bigcap_{j\ge i}P_i \subseteq \bigcap_{i}\bigcup_{j\ge i}P_i. \tag{*} $$
Now consider an arbitrary poset $L$ which forms a complete lattice $(L, \le)$, and define
$$ \mathop{\lim\sup}\limits_{n\to\infty}(x_n) = \inf\limits_{m\ge0}\sup\limits_{n \ge m}(x_n)\\ \mathop{\lim\inf}\limits_{n\to\infty}(x_n) = \sup\limits_{m\ge0}\inf\limits_{n \ge m}(x_n), $$
where $x_i \in L$. Under what conditions on $(L, \le)$ and $(x_n)$, if any, is it true that
$$ \mathop{\lim\inf}\limits_{n\to\infty}(x_n) \le \mathop{\lim\sup}\limits_{n\to\infty}(x_n)? \tag{**} $$
For instance, $(*)$ seems to be an instance of $(**)$ for the case that $L$ is the powerset $2^X$ of some base set $X$, with set inclusion as the partial order.