# If $\sum (a_n)^2$ converges and $\sum (b_n)^2$ converges, does $\sum (a_n)(b_n)$ converge?

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.

• Hint: Cauchy-Schwartz inequality. May 13, 2015 at 2:31
• using cauchy will require knowing either 1) $l^2$ is an inner product space or 2) monotone convergence for real numbers, which seems slightly complicated for such a simple problem (not to say I don't approve!)
– cats
May 13, 2015 at 2:36
• Vote to close. The OP has not returned for 2 years. Apr 28, 2021 at 17:16
• @cats how is monotone convergence complicated for any problem? Without there's no comparison test which is the simplest of the convergence tests Feb 22, 2023 at 9:09

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.
• But how do I know that $(a_n)(b_n)\ge0$ so that I can apply the comparison test? Can I really try sequences just like numbers? thanks May 13, 2015 at 2:36
• Easy enough fix, but you might need to specify that $a_n$ and $b_n$ are positive or else take the absolute value and then apply the above result to show it absolutely converges and therefore convereges