Given: is closed under .
Show S is closed and unbounded in .
If S were not closed, then . Thus, S must be closed since sup(A) must be in S or equal to S (by the definition of closed).
If S were bounded, then . Choose , then there does not exist , since in this case. Thus, a contradiction and S must be unbounded. Therefore, S is closed and unbounded.
NOW, the problem... The second part dealing with bounded does not cover the case where there is no that has . So, I need help showing S is unbounded for this possibility. Any takers on helping? Let me know if something is unclear.