Proof of Cauchy-Schwarz Inequality

[barksdalemc]

I have a homework problem in which I have to prove the Cauchy-Schwarz inequality. I tried to do it by induction, but when I try to do summation to 2, I get a mess of terms. The professor hinted that one can use the fact that geometric means are less than or equal the arithmetic mean, but I can't seem get past just that hint. Should I continue trying induction or am going down the wrong road?

 

[StatusX]

The slick way is to look at the inner product of x+a*y with itself, where x,y are vectors and a is a scalar. This must be positive for all a, and picking the right a will give you the result.