Convergence of Series: $\sum a_nb_n$

[Krizalid1]

Let $a_n$ and $b_n$ be sequences in $\mathbb R.$ Show that if $\displaystyle\sum b_n$ converges and $\displaystyle\sum|a_n-a_{n+1}|<\infty,$ then $\displaystyle\sum a_nb_n$ converges.

 

[Aaron Bex]

My attempt:
Let $S_n=\displaystyle\sum_{k=1}^{n}a_nb_n.$ We have $$|S_n-S_{n-1}|=|a_nb_n|\leq|b_n||a_n-a_{n+1}|.$$ Since $\displaystyle\sum|b_n|$ converges and $\displaystyle\sum|a_n-a_{n+1}|<\infty,$ then it follows that
$\displaystyle\sum|S_n-S_{n-1}|<\infty.$
By Cauchy's convergence criterion, it follows that the sequence $\left(S_n\right)$ is convergent, which implies that $\displaystyle\sum a_nb_n$ converges.
Is my proof correct? If no, then how can I correctly prove the statement?A:

Your proof is not quite valid as it stands since you need to show that $S_n$ is a Cauchy sequence, not merely that its differences are bounded.

Let $(S_n)$ denote the partial sums of $\sum a_n b_n$, i.e. $S_n = \sum_{k=1}^n a_k b_k$. Then for any $n \ge