If a,b,c are three positive numbers then using cauchy schwerz inequality prove that

$(a+b+c)(\frac1a+\frac1b+\frac1c) \geq 3^2$

Consider two vectors

$\begin{align} u &= ( \sqrt a , \sqrt b, \sqrt c )\\ v &= (\frac{1}{\sqrt a} , \frac{1}{\sqrt b}, \frac{1}{\sqrt c})\\ u.v &= ( \sqrt a , \sqrt b, \sqrt c ) (\frac{1}{\sqrt a} , \frac{1}{\sqrt b}, \frac{1}{\sqrt c})\\ u.v &= (\frac{ \sqrt a}{\sqrt a} +\frac{\sqrt b}{\sqrt …