Suppose that S is a nonempty set of real numbers and that theta is an upper bound of S. Prove that the following conditions are equivalent:
(1) We have theta = Sup S.
(2) There exists a sequence (X sub n) in S such that (X sub n) approaches theta as n approaches infinity.
I agree with Drexel. If you take a bounded sequence with a maximum and an infimum, then (1) does not imply (2).
In other words, if you choose a sequence such that you'll ever get sufficiently close to , the sequence cannot continue on until infinity. Or, if you choose a sequence that will continue on until infinity, it will always reside in the -neighborhood of L.