## One line proof of CS inequality

I don’t think I have seen shorter proof of Cauchy-Schwarz Inequality than the following

$\displaystyle\frac{\sum a_k b_k}{\sqrt{(\sum a_{k}^{2})(\sum b_{k}^{2})}}=1-\frac{1}{2}\sum\left(\frac{a_k}{\sqrt{(\sum a_{k}^2)}}-\frac{b_k}{\sqrt{(\sum b_{k}^2)}}\right)^2$

which immediately gives

$\displaystyle\frac{\sum a_k b_k}{\sqrt{(\sum a_{k}^{2})(\sum b_{k}^{2})}}\le 1$

from which CS inequality follows.

Source: Mathematics Magazine, Vol. 68, No. 2 (Apr., 1995), p. 97