Convergent series question - Dirichlet's Test? 
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.

Hint:
Let $S_n= \sum_{k=1}^{n}b_n$ and construct the partial sum
$$ V_n = \sum_{k=0}^{n}a_n b_n = a_1 b_1 + a_2 b_2 + \dots + a_n b_n $$
$$ = a_1 S_1 + a_2 (S_2-S_1) + \dots + a_n (S_n-S_{n-1})  $$
$$ = S_1(a_1-a_2)+ S_2(a_2-a_3)+ \dots + S_{n-1}(a_{n-1}-a_n)+ a_n S_n $$

$$ \implies V_n= \sum _{k=1}^{n-1}S_k (a_k-a_{k-1}) + a_n S_n .$$

