# Math Help - Cauchy–Schwarz inequality- Question

1. ## Cauchy–Schwarz inequality- Question

Help with that

2. Originally Posted by sbsite
Help with that
Simple, since $X,Y,Z,W>0$

$\bold{a}=<\sqrt{X},\sqrt{Y},\sqrt{Z},\sqrt{W}>$

$\bold{b}=\left< \frac{1}{\sqrt{X}},\frac{1}{\sqrt{Y}},\frac{1}{\sq rt{Z}},\frac{1}{\sqrt{W}} \right>$

Thus the dot product is,
$\bold{a}\cdot \bold{b}=1+1+1+1=4$

Thus,
$4\leq ||\bold{a}||||\bold{b}||$
Square both sides,
$16\leq (X+Y+Z+W)\left(\frac{1}{X}+\frac{1}{Y}+\frac{1}{Z} +\frac{1}{W} \right)$

3. this is also a result of

$(a_1+a_2+\dots+a_n)\left(\frac{1}{a_1}+\frac{1}{a_ 2}+\dots+\frac{1}{a_n}\right)\ge n^2$

assuming that $a_i>0$ for all $i$ such that $1\le{i}\le{n}$

that can be proved by either using AM-GM or Cauchy