How does $(x-y)^2 \geq 0$ imply $xy ≤ x^2/2 + y^2/2$?

Question

I am currently reading the book "The Cauchy-Schwarz Master Class" and they give us the equality $(x-y)^2 \geq 0$ and say that this imply $xy ≤ x^2/2 + y^2/2$

I am having trouble finding out exactly why this is the case. I can plot numbers into the inequality and I can see that it holds, but why does this one equality imply the other?

Answer 1

Just expand!

$$(x-y)^2 \geq 0 \Rightarrow x^2 + y^2 - 2xy > 0 \Rightarrow x^2 + y^2 \geq 2xy \Rightarrow \frac{x^2}{2} + \frac{y^2}{2} \geq xy$$

Answer 2

$xy ≤ x^2/2 + y^2/2 \iff 2xy \le x^2+y^2 \iff 0 \le x^2-2xy+y^2 =(x-y)^2$.