The whole point is that may not be a covergent sequence.
Take a close look at the example that I gave above.
In that example the sequence does not converge. Now it is bounded.
So from the given, you may not conclude that the sequence converges because there is a counterexample.
How do you prove that it is bounded?
the given was and not alone..
you should not put your interest with since what we want is the sequence and that sequence is different from !
if you find any "flaws" with my "proof" while i had used only the given and the definition and some theorems, point it out.. well, i would try to consult this with my professor tomorrow..