Proposition Suppose $\{a_n\}$ and $\{b_n\}$ are real sequences such that $a_n \to 0$. Show that $\sum a_kb_k$ converges under the following two conditions: $\sum_{k=1}^{\infty} |a_{k+1}-a_k|< \infty$ and $|\sum_{k=1}^{n} b_k| \leq M$.
I'm stuck on this proof because I'm not sure how this: $\sum_{k=1}^{\infty} |a_{k+1}-a_k|< \infty$ fits into the bigger picture. In terms of finite sums this would be a telescoping series.