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?
Theorem: A convergent sequence is bounded.. (do i need to prove this?)
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..