Let {xn}n=1∞ be a bounded sequence.
a) Prove that there exists an s such that for every r>s, there exists an M∈N such that for all n≥M, we have xn<r.
b) If s is a number as in a), then prove n→∞limsupxn≤s.
c) Show that if S is the set of all s as in a), then n→∞limsupxn=infS.