For sequences of real numbers, it's defined as the least upper bound (sup)/greatest lower bound(inf). For sequences of sets, it's defined as the set of elements that occur in infinitely many of the sets (sup)/all but finitely many of the sets(inf).
Is there a way to relate these two definitions, conceptually? I'm trying to think of a way to make it apparent that lim sup/inf is really just one concept, and not just two separate definitions for sets/sequences that have no relationship. So far, I've tried thinking of a vague "smallest bounding" set for sequence of sets.
The idea that lim inf of a sequence of sets is subset of the lim sup also throws me off, as it seems like there's no corresponding relationship in the definition for real sequences.