Multiplicating inequalities

You can multiply these inequalities, because both sides of both inequalities are nonegative. Here is an explaination:

$$|x| \leq \sqrt{x^2+y^2}$$

$|y| \geq 0$, so you can multiply both sides by $|y|$:

$$|xy| \leq |y|\sqrt{x^2+y^2}$$

But we know that $|y| \leq \sqrt{x^2+y^2}$ and $0 \leq \sqrt{x^2+y^2}$, so:

$$|y|\sqrt{x^2+y^2} \leq \sqrt{x^2+y^2} \times \sqrt{x^2+y^2}=x^2+y^2$$

Finally:

$$|xy| \leq |y|\sqrt{x^2+y^2} \leq x^2+y^2$$