Proof that $|x|+|y|\leq\sqrt{2(x^2+y^2)}$ How do I prove that for $x,y\in\mathbb{R}$ we have $|x|+|y|\leq\sqrt{2(x^2+y^2)}$?
I thought that $(|x|+|y|)^2=x^2+y^2+2|x||y|\leq2(x^2+y^2)$, but I'm not sure why that holds.

The claim immediately follows from
$$\bigl(|x|+|y|\bigr)^2\leq\bigl(|x|+|y|\bigr)^2+\bigl(|x|-|y|\bigr)^2=2\bigl(|x|^2+|y|^2\bigr)\ .$$
