Sigma-algebras and measures

Let $\mu$ be a finite measure and $A_n$ measurable with $\sum\mu(A_n) < \infty$. (a) Show $\mu(\bigcup_{k\ge n}A_k)\to0$. (b) Deduce that $\mu(\{x : x\in A_n\text{ for infinitely many }n\}) = 0$ (the Borel–Cantelli lemma). (c) Does (b) need $\mu$ finite?