I missed the last class and I guess our teacher solved this problem and said it would be on our next quiz and I do not really understand how to do it.
(i) lim sup sn is an element of SL(sn)
n to infinity
(ii) same thing but replace the sup with inf
lim sup:= limit supremum
lim inf:= limit infimum
The Attempt at a Solution
I know that it wants me to prove that the limit supremum of a sn is an element of the set of all limits of all convergent subsequences of the sequence sn. It makes sense that the sup and inf of the sequence would be would be elements of the set of all convergent subsequences because in order to have convergent subsequences the original sequence must be bounded. Thus, bounds would seem to be elements of the convergent, bounded subsequences.