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ThePerfectHacker · #7 June 25th, 2007, 02:59 PM

Quote:

Originally Posted by shosho

could you please explain in another way

How about this way?

We need to use the inequality:
$x^2+y^2+z^2 \geq xy+yz+xz$
For all real numbers $x,y,z>0$ .

Begin by noticing that, by the AM-GM inequality,
$x^3+y^3+z^3 \geq 3xyz$
Thus,
$x^3+y^3+z^3 - 3xyz \geq 0$
Factor,
$(x+y+z)(x^2+y^2+z^2 - xy - yz - xz) \geq 0$
Since $(x+y+z)>0$ we can cancel to obtain,
$x^2+y^2+z^2 \geq xy+yz+xz$

Which is the Cauchy-Swartz inequality.