I guess is a sequence of real numbers. Well, if is unbounded, then the sequence is either unbounded below or unbounded above. In the first case, we can find a subsequence which converges to , and in the second case we can find a subsequence which converges to . Here, since in the first part, then is not bounded above.
If we have for a subsequence of , it doesn't mean that . For example, take (the limit doesn't exist).