Use this well known inequality: $\displaystyle \left| {\left| x \right| - \left| y \right|} \right| \leqslant \left| {x - y} \right|$.
Part (a) follows at once.
For part (b), consider $\displaystyle \left( {b_n } \right) = \left( { - 1} \right)^n $ .
I just wanted to say thanks for responding to my question. Luckily I was able to locate the answers you listed in my notes from class. Unfortunately I stayed up all night completing this assignment.