$\sum a_n$ converges conditionally and $\sum b_n$ converges absolutely, then will $\sum a_nb_n$ converge absolutely?

Question

Suppose $\sum a_n$ converges conditionally and $\sum b_n$ converges absolutely, then will $\sum a_nb_n$ converge absolutely?

I know that $\sum|a_n|$ does not converge while $\sum a_n$ does and that $\sum|b_n|$ does converge, and also that $\lim_{n \to \infty} |a_nb_n|=0$ but I'm not sure how to proceed from here.

Answer 1

$|a_{n}|<1$ eventually and so $|a_{n}b_{n}|\leq|b_{n}|$ eventually, but $\displaystyle\sum|b_{n}|<\infty$, then so is $\displaystyle\sum|a_{n}b_{n}|<\infty$.