TITLE: Is $\sum\limits_{n=1} ^\infty |x_n||y_n| ≤\left(\sum\limits_{n=1} ^\infty |x_n|\right)\left(\sum\limits_{n=1} ^\infty |y_n|\right)$?
QUESTION [7 upvotes]: Is $\sum\limits_{n=1} ^\infty |x_n||y_n| ≤ \left(\sum\limits_{n=1} ^\infty |x_n|\right)\left(\sum\limits_{n=1} ^\infty |y_n|\right)$?
I want to use this in a separate exercise but I can't think if this is true let alone how to prove it and I can't find any resources to help.

REPLY [2 votes]: How about something more elementary?
If $x_i=0$ for all $i$, it is easy.  So assume not.
Let $M = \max\{|y_1|,|y_2|,\dots\}$.  If $M=\infty$ it is easy, so assume not.
Certainly $\sum_{i=1}^\infty |y_i| \ge M$.
For all $i$, we have $|x_i y_i| \le |x_i|\; M$, so
$$
\sum_{i=1}^\infty |x_i y_i| \le \sum_{i=1}^\infty \big(|x_i|M\big) =
\left(\sum_{i=1}^\infty |x_i|\right)M \le
\sum_{i=1}^\infty |x_i| \sum_{i=1}^\infty |y_i|
$$