$x,y,z$ lập thành CSN $\Leftrightarrow xz=y^2$. Ta có
$\frac{1}{3}(x+y+z)=\frac{1}{3}(x+\sqrt{xz}+z)$
$\sqrt[3]{xyz}=\sqrt[3]{y^3}=y=\sqrt{xz}$
$ \sqrt{\frac{1}{3}(xy+yz+zx)} =\sqrt{\frac{1}{3}(xz+z\sqrt{xz}+x\sqrt{xz})}$
Ta thấy
$\frac{1}{3}(x+y+z). \sqrt[3]{xyz} = \frac{1}{3}(x+\sqrt{xz}+z).\sqrt{xz}=\frac{1}{3}(xz+\sqrt{xz}z+\sqrt{xz}x)=\frac{1}{3}(xy+yz+zx)$
Suy ra đpcm.