How do we prove that $\frac{x^{2} + y^{2}}{2}\geq \sqrt{2(x-y)^{2}}$ if $xy = 1$ and $x\neq y$?

Question

Apparently this can be solved using AGM:

$$(xy = 1)\wedge(x \neq y) \Rightarrow \frac{x^{2} + y^{2}}{2}\geq \sqrt{2(x-y)^{2}}$$

I've tried doing

$$\frac{x^2 + y^2}{2} \geq \sqrt{x^2 y^2}$$

Answer 1

Hint:

$$\frac{x^{2} + y^{2}}{2} = {(x-y)^2+2\over 2} $$

Now you can apply your idea.