<s>[INST] How to prove $2x^2 + 2y^2 + 2z^2 -2xy -2yz \geq 0$ I know how to prove $2x^2 + 2y^2 + 2z^2 -2xy -2yz -2xz \geq 0$ since I can split it up in the three terms $(x-y)^2, (y-z)^2, (z-x)^2$ which are all greater than or equal to zero, but how do I prove this without the last term: $2x^2 + 2y^2 + 2z^2 -2xy -2yz \geq 0$? 
 [/INST] It is $$x^2+y^2-2xy+y^2+z^2-2yz+z^2+x^2\geq 0$$
 so
$$(x-y)^2+(y-z)^2+z^2+x^2\geq0$$
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