Prove that for all real numbers $x,y$ and $z$ that $x^2+y^2+z^2 \geq x\sqrt{y^2+z^2}+y\sqrt{x^2+z^2}$ 
Prove that for all real numbers $x,y$ and $z$ that $x^2+y^2+z^2 \geq x\sqrt{y^2+z^2}+y\sqrt{x^2+z^2}$.

It looks like AM-GM should be used here but the square roots make it difficult. So maybe Cauchy-Schwarz works?

This is simply 2 applications of $ab\leq\frac{1}{2}(a^2+b^2)$:
$$
x\sqrt{y^2+z^2}\leq\frac{1}{2}(x^2+y^2+z^2),\quad y\sqrt{x^2+z^2}\leq\frac{1}{2}(y^2+x^2+z^2)
$$
Now, sum the above 2 inequalities.
