Question

Prove that if the sequence (un) is convergent and bounded above by M, then the limit is
bounded above my M.
[Hint: Assume that the limit is larger than M and show that a contradiction arises when
 is suitably chosen in the definition of limit.]

Answer 1

If the limit is larger than M, there exists un (n-> ∞) , such that |un| > M, so the sequense is not bounded by M.