What have you tried? It looks like (a) is fairly straightforward. In fact, by the result of part (b), it appears that you should be able to show that $|f| \le 1$, $\mu$-a.e. How do you show that something is true $\mu$-a.e.? Start with "Given $\varepsilon>0$, ..." and end with the $\mu$-measure of the set on which it is not true is less than $\varepsilon$.
Hint: let $\displaystyle A_k = \bigcup_{n \ge k} c_n$. What can you say about $\displaystyle \int \left|f-\chi_{A_k}\right|d\mu$ as $k \to \infty$?