I think is an easy problem, but I'm confused, the statement establishes that is bounded below, but not above.
How to prove this?
Which part are you having trouble with? Try firstly the only if statement. Then, you want to prove that but evidently by assumption you have that for every so that from where the conclusion follows since were arbitrary (intuitively you can take the limit as of both sides of this last expression, but the left side is 'unaffected' by the limit since there is no )