TITLE: If $\sum_{n=1}^\infty{a_n}$ and $\sum_{n=1}^\infty{b_n}$ converge then If $\sum_{n=1}^\infty{a_n*b_n}$ converge
QUESTION [5 upvotes]: My attempt; 
Since $a_n$ converges then its sequence of partial sums $(x_n)$ converges to a limit x. Also, since $b_n$ converges then its sequence of partial sums $(y_n)$ converges to a limit y. Observe that by using the multiplication limit theorem:
If $x_n \to x$ and $y_n \to y$ then $x_n*y_n \to x*y$. We know that the product of its partials sums converge, thus the product of the series converges.$
Is this correct? Any guidance is appreciated !

REPLY [2 votes]: This isn't true in general. there is a easy conterexample:
Regard $a_n =b_n= \frac{(-1)^n}{\sqrt{n}}$ then $a_nb_n = \frac{1}{n}$.
you can use Leibniz criterion to check the convergence of $\sum^\infty_{n=1} a_n$ but $\sum^\infty_{n=1} a_nb_n = \sum^\infty_{n=1} \frac{1}{n}$ will not converge!
if you assume a little bit more you can fix that!
Either if one is $a_n$ or $b_n$ is monoton then you can  dirichlets test or if one converge absolute then you can use holder inequality:
$\left|\sum_{n=1}^\infty a_n b_n\right| \leq \sup|a_n|\sum_{n=1}^\infty |b_n|$