Sorry, not a specialist in this area, but what is Sn? Is it just a sequence of real numbers S1, S2, ...? If so, then what is sup Sn?
I need to prove
i) lim n to infinity sup Sn is an element of SL(Sn)
ii) same thing but replace sup with inf
I know SL(Sn) is the set of all limits of all convergent subsequences of Sn.
Also, lim n to infinity sup Sn can be defined as sup SL(Sn)
So it may be easier to show that Sup SL(Sn) is an element of SL(Sn)
Let's prove that there exists a subsequence whose limit is . Let and let . We can ensure that for every and every , , i.e., the entire tail of the subsequence starting from lies in the -neighborhood of .
Indeed, suppose is given and have already been chosen. From the definition of , there is a point such that
for all , (*)
Without loss of generality, we can assume that ; otherwise, choose a new as ; then (*) still holds. In particular, , i.e., . From the definition of sup, there exists an index, which we'll call , such that and . Thus, .
TL;DR: We choose a subsequence whose limit is the limit superior of (call it ). Since is a limit, sups of tails of come close to . Since those are sups, there are individual elements that come close to .