Young's inequality for products

$$\forall \epsilon >0,x,y\in \mathbb{R}:2|xy|\le \epsilon x^2+\frac{1}{\epsilon }{y}^2$$

Proof:

${\left(\sqrt{\epsilon }|x|-\frac{1}{\sqrt{\epsilon }}|y|\right)}^2\ge 0$ (left is positive) $⟹\epsilon |x{|}^2+\frac{1}{\epsilon }|y{|}^2\ge 2|x||y|$