Me noticed that, you can generalize limits of sequences of vectors of real numbers and limits of subsets of a set to what i've called $\mathbb{N}$-complete lattices. Simply, given such lattice $P$ and $u_k: \mathbb{N} \to P$ define $$\lim_k \sup u_k := \inf_{n \in \mathbb{N}} \sup_{k\geq n} u_k$$
And define $\lim \inf$ analogously; then limit exists iff $\lim \sup = \lim \inf$ and it's equal to the common value. This is pretty trivial (if i didn't do mistake), so i'm surprised i didn't find a standard name for this kind of lattices. A lattice is a poset with finite suprema and infima; a complete lattice is a poset with arbitrary suprema and infima. Is there a standard name for inbetweens like this one?
(Btw, complete lattices are not equivalent to my lattices: just consider the smallest uncountable ordinal, it's clearly the latter but not the former).
These are called countably complete lattices. More generally, a lattice is $\kappa$-complete iff every subset of size $<\kappa$ has a supremum and infimum (so countable completeness is $\omega_1$-completeness and all lattices are trivially $\omega$-complete).
We can also "mix-and-match" levels of completeness; e.g. a lattice could be $\omega_2$-upper-complete but only $\omega_1$-downward-complete (all size-$<\omega_2$ sets have sups but only countable sets are guaranteed to have infs). This comes up less frequently, however, as far as I'm aware.