Let {sn} be a sequence and let E be the set containing members x such that x is a limit of a subsequence of sn. Let S*=sup(E).

Prove S* is a member of E.

I do not understand the first part of the proof (from a textbook). It goes if S* = infinity then E is unbounded (fine so far). Then {sn} is unbounded, hence there is a subsequence -> infinity.

This raised a few questions.

If {sn} is unbounded, does sn -> + or - infinity?

The proof also seems to imply {sn} -> infinity iff xn -> infinity, where xn is a subsequence. Correct?

Thanks