to show $\sum_{i=1}^{\infty} |x_i y_i|$ converges

Question

$X$ consists of sets of the form $(x_1, x_2, x_3, \dots)$ where $x_i \in \mathbb R$.

Suppose $\sum_{i=1}^{\infty} x_i ^2$ converges.

Show that : $\sum_{i=1}^{\infty} |x_i y_i$| converges. where $x,y \in X$

I am able to show that $|\sum_{i=1}^{\infty} x_i y_i| \le [\sum_{i=1}^{\infty} x_i^2]^\frac12 *[\sum_{i=1}^{\infty} y_i^2]^\frac12$ (using cauchy schwarz)

So $|\sum_{i=1}^{\infty} x_i y_i|$ converges. But how to bring mod sign inside $\sum$ so that I can show that $\sum_{i=1}^{\infty} |x_i y_i|$ converges .

Answer 1

Cauchy-Schwarz gives that $\sum_1^n |x_i||y_i|\le (\sum_{i=1}^n x_i^2)^{1/2}(\sum_{i=1}^n y_i^2)^{1/2}$. And absolute convergence implies convergence.