$\sum a_n $ absolutely convergent and $\sum b_n $ convergent $\implies \sum a_nb_n$ absolutely convergent. Is this true?
$\sum a_n $ absolutely convergent and $\sum b_n $ convergent $\implies \sum a_nb_n$ absolutely convergent.
I don't know how to proceed .Please help.

Other way:
If $\sum a_n$ and $\sum b_n$ absolutely converge, then $\sum\sqrt{|a_nb_n|}$ converges, as
$$0\le2\sqrt{|a_nb_n|}\le |a_n|+|b_n|$$
And $|a_nb_n|<1$ for $n$ large and so
$$|a_nb_n|<\sqrt{|a_nb_n|}$$
