$\sum a_n$ 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?

Basically you can use this fact

For sure, if $x_n>0$ and $\sum x_n$ converges then exists $n_0\in\Bbb N$ such that $n>n_0\implies x_n<1\implies x_n^2<x_n$. Then by comparison $\sum x_n^2$ converges.

If $a_n,b_n>0$ are such $\sum a_n$ and $\sum b_n$ converges, then $\sum a_n^2$ and $\sum b_n^2$ converges as well. Then,
$$(a_n-b_n)^2\geq0\implies a_n^2+b_n^2\geq2a_nb_n,\quad\forall n\in\Bbb N.$$
Again, by comparison, have that $\sum a_nb_n$ converges.
