What is wrong in my solution for (if $\Sigma a_n$ converges and $\{b_n\}$ is bounded and monotonic, then $\Sigma a_nb_n$ converges)?

Question

Since $\{b_n\}$ is bounded, $b_n\leq M$, then

$\Sigma a_nb_n\leq M\Sigma a_n$, and so converges.

I did not use that $b_n$ is monotonic.

Answer 1

$$\Sigma a_nb_n\leq M\Sigma a_n$$ is not true unless $ a_n \ge 0$

Answer 2

Consider $a_n=\frac{(-1)^n}{n}$ and $b_n=(-1)^n$.