$\sum a_n$ and $\sum b_n$ converges. Prove $\sum a_nb_n$ converges. Let ${a_n}$ and ${b_n}$ sequences with positive terms, such that $\sum a_n$ and $\sum b_n$ converges. Prove that $\sum a_nb_n$ converges as well.
What I did:
$\sum a_nb_n \le \left(\sum a_n\right) \left(\sum b_n\right)$ and by comparison test we are done.
However, when I asked my teacher regarding that solution, he told me that I can assume that inequality for finite numbers but for infinite numbers, it is not valid. It seems to me that $\left(\sum a_n\right) \left(\sum b_n\right)$ is absolute converges so that are no Riemann issues with this.
Can any one enlighten me and explain why is this invalid?

Since $\sum\limits_{k = 1}^\infty b_k$ converges, then $b_n \to 0 \ (n \to \infty)$. Note that $\{b_n\}$ is a sequence of positive numbers, thus there exists $N \in \mathbb{N}_+$ such that$$
0 < b_n < 1. \quad \forall n > N
$$
Therefore,$$
0 < \sum_{k = 1}^\infty a_k b_k = \sum_{k = 1}^N a_k b_k + \sum_{k = N + 1}^\infty a_k b_k \leqslant \sum_{k = 1}^N a_k b_k + \sum_{k = N + 1}^\infty a_k < +\infty.
$$
