If $\sum (a_n)^2$ converges and $\sum (b_n)^2$ converges, does $\sum (a_n)(b_n)$ converge?
Could someone help me to solve this or at least give me a hint?, I have tried using Cauchy's criterion, the Dirichlet test for convergence, etc, but I can´t prove it.Honestly I don´t know where to start. Any help will be appreciated.
Best Answer
Start from here :$$(|a_n|-|b_n|)^2=a_n^2+b_n^2-2|a_nb_n|\ge 0$$
$$\implies |a_nb_n|\le \frac{1}{2}(a_n^2+b_n^2)$$By comparison test, $\sum a_nb_n$ is absolutely convergent , hence convergent.