Prove the inequality $2|xy|≤x^2+y^2$

Question

As far as I know, there are no special conditions other than this is a space with only real numbers. I'm not sure how to move forward with the problem. I know that $x^2 = |x|^2$ but I don't know how to make use of that.

$2|xy|≤ x^2 + y^2$

Answer 1

Note that $$0\leq (|x|-|y|)^2 = x^2 -2|xy| +y^2.$$

Answer 2

By AM-GM $$x^2+y^2\geq2\sqrt{x^2y^2}=2|xy|.$$

Also, we can use the Rearrangement: $$a^2+b^2=|a|^2+|b|^2\geq |a||b|+|b||a|=2|ab|.$$

Answer 3

$$x^2+y^2-2|xy|=\min(x^2+y^2-2xy,x^2+y^2+2xy)\ge0.$$