x+yz=(1-y)(1-z)
y+zx=(1-z)(1-x)
z+xy=(1-x)(1-y)

由柯西不等式
{sqrt[(1-y)(1-z)] +sqrt[(1-z)(1-x)] + sqrt[(1-x)(1-y)] }^2
<=[(1-y)+(1-z)+(1-x)][(1-z)+ (1-x) + (1-y)] =4
于是原不等式得证。