Limits Superior and Inferior
As we look at sequences (and nets) of real numbers (and more general ordered spaces) a little more closely, we’ll occasionally need the finer notion of a “limit superior” (“limit inferior”). This is essentially the largest (smallest) value that a sequence takes in its tail.
In general, let be a net (indexed by ) in some ordered space . Then we can consider the “tail” of the index set consisting of all indices above a given index . We then ask what the least upper bound of the net is on this tail: . Alternately, we consider the greatest lower bound on the tail: .
Now as we move to tails further and further out in the net, the least upper bound (greatest lower bound) may drop (rise) as we pass maxima (minima). That is, the supremum (infimum) of a set bounds the suprema (infima) of its subsets. So? So if we pass such a maximum it clearly doesn’t affect the long-run behavior of the net, and we want to forget it. So we’ll take the lowest of the suprema of tails (the highest of the infima of tails).
Thus we finally come to defining the limit superior
and the limit inferior
Now these are related to our usual concept of a limit. First of all,
and the limit converges if and only if these two are both finite and equal. In this case, the limit is this common finite value. If they both go to infinity, the limit diverges to infinity, and similarly for negative infinity. If they’re not equal, then the limit bounces around between the two values.
If we’re considering a sequence of real numbers, then we’re taking a bunch of infima and suprema, all of which are guaranteed to exist. Thus the limits superior and inferior of any sequence must always exist.
As an illustrative example, work out the limits superior and inferior of the sequence . Show that this sequence diverges, but does so by oscillating rather than by blowing up.
Finally, note that we can consider a function defined on a ray to be a net on that ray, considered as a directed subset of real numbers. Then we get limits superior and inferior as goes to infinity, just as for sequences.