Proofs on inequalities of real numbers

Question

So I have these inequalities (statements) to prove:

$x,y \in \mathbb{R}$

$\vert xy \vert \leq \frac{1}{2}(x^2 + y^2)$

$x,y \geq 0 \implies xy \leq \frac{1}{4}(x + y)^2$

I know that I have to use "The Axioms of the real numbers" but I don't have any idea how to do it.

Answer 1

Use that $$(|x|-|y|)^2=x^2+y^2-2|xy|$$