I proved this way:
Since {$b_n$} is bounded, let $|b_n| \le \alpha$ and $\alpha$ is an upper bound.
Since $\sum a_n$ converges, there exists N such that $|\sum_{k=m}^{n} a_k| \le \frac{\epsilon}{\alpha}$ for every N $\le$ n,m.
Then $|\sum_{k=m}^{n} a_k \alpha| \le \epsilon$ for every N $\le$ n,m.
Now, $|\sum_{k=m}^{n} a_k b_k| \le |\sum_{k=m}^{n} a_k \alpha| \le \epsilon$, thus $\sum a_n b_n$ converges.
But the professor said the last inequality is wrong. What is the problem? How to fix it?