Minkowski space-time of special relativity (4-D)

[sana307]

In Minkowski space-time of special relativity (4-D, time with R^3),

I want to show that:
If x is time like (i.e <x,x> < 0 (inner product))
<x,x> * <y,y> <= (<x,y>)^2 for any y

note: <x,x> = -(x_0)^2 + (x_1)^2 + (x_1)^2 + (x_1)^2

Please help me. I already spent hours with it.
I try to solve it by follow the proof of Cauchy Schwarz inequality,
but I can't not get it.
I realize that in the proof of Cauchy Schwarz inequality,
it always required <x,x) >=0.

Thank you very much!