Convergence of series $\sum a_rb_r$

Question

Assuming $\sum a^2_r$ and $\sum b^2_r$ converge, can we deduce that $\sum a_rb_r$ converges? It feels like we can, but how? Using Cauchy Criterion for convergence maybe?

Can you hint me?

Thanks a lot!

Answer 1

$$2\sum_r|a_rb_r|\leqslant\sum_ra_r^2+\sum_rb_r^2$$

Answer 2

Apply Cauchy-Schwarz inequality to get $$\sum|a_rb_r|\le \sqrt{\sum a_r^2\sum b_r^2}$$ So the series is absolutely convergent and hence convergent.