a) Derive the inequality
$\left|X\right|\left|Y\right|-X\cdot Y\le\left|X-Y\right|^2$
(Hint: Show that $\left|X\right|\left|Y\right|\le\frac{1}{2}\left(X^2+Y^2\right)$and add $\frac{1}{2}\left|X-Y\right|^2$to the right side of this last inequality.)
(b) Use the result of part (a) to verify that
$\left|\frac{X}{\left|X\right|}-\frac{Y}{\left|Y\right|}\right|\le\frac{\sqrt2}{\sqrt{\left|X\right|\left|Y\right|}}\left|X-Y\right|$
How do I use Cauchy inequality to solve this inequality?
I have no idea.