Real Analysis Answers

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.]

Prove that if the sequence (an) is convergent with the limit 0, and the sequence (bn) is bounded, then the sequence (an bn ) is convergent with limit 0