Áp dụng bất đẳng thức cosi cho 2 số dương
\(\dfrac{1-x+1-z}{2}\ge\sqrt{\left(1-x\right)\left(1-z\right)}\Leftrightarrow\left(1-x\right)\left(1-z\right)\le\left(\dfrac{x+y+z-x+x+y+z-z}{2}\right)^2\Leftrightarrow\left(1-x\right)\left(1-z\right)\le\dfrac{\left(x+y+z+y\right)^2}{4}\Leftrightarrow4\left(1-x\right)\left(1-z\right)\le\left(1+y\right)^2\Leftrightarrow4\left(1-x\right)\left(1-y\right)\left(1-z\right)\le\left(1+y\right)^2\left(1-y\right)\Leftrightarrow4\left(1-x\right)\left(1-y\right)\left(1-z\right)\le\left(1+y\right)\left(1-y^2\right)=\left(x+y+z+y\right)\left(1-y^2\right)=\left(x+2y+z\right)\left(1-y^2\right)\Leftrightarrow4\left(1-x\right)\left(1-y\right)\left(1-z\right)\le x+2y+z\)
(vì y<1\(\Rightarrow y^2< 1\))
