Square both sides inequality help

[evagelos]

Homework Statement

Prove: xyzw$$\geq x+y+z+w-3$$,where $$x\geq 1,y\geq 1,z\geq 1,w\geq 1$$

Homework Equations

The Attempt at a Solution

The only thing i could try is to square both sides but then this leads nowhere.

Any ideas??

 

[Tedjn]

Start with the following inequalities: (Why are they true?)

0 ≤ (xy-1)(zw-1)
0 ≤ (x-1)(y-1)
0 ≤ (z-1)(w-1)​

 

[evagelos]

thanks!