VTRMC 2006 Problem 5

Let $\{a_n\}$ be a monotonically decreasing sequence of positive real numbers with limit $0$ . Let $\{b_n\}$ be a rearrangement of the sequence such that for every non-negative integer $m$ , the terms $b_{3m+1}$ , $b_{3m+2}$ , $b_{3m+3}$ are a rearrangement of the terms $a_{3m+1}$ , $a_{3m+2}$ , $a_{3m+3}$ . Prove or give a counterexample to the following statement: the series $\sum_{n=1}^\infty(-1)^nb_n$ is convergent.