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)