Multiplicating inequalities I have two inequalities: $|x|\leq\sqrt{x^2+y^2}$ and $|y|\leq\sqrt{x^2+y^2}, \forall x,y \in \Bbb R$, can I multiply these inequalities to get $|xy|\leq x^2+y^2$?
If yes, what is the justification? If not, why?

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