Help with that
Simple, since $\displaystyle X,Y,Z,W>0$
$\displaystyle \bold{a}=<\sqrt{X},\sqrt{Y},\sqrt{Z},\sqrt{W}>$
$\displaystyle \bold{b}=\left< \frac{1}{\sqrt{X}},\frac{1}{\sqrt{Y}},\frac{1}{\sq rt{Z}},\frac{1}{\sqrt{W}} \right>$
Thus the dot product is,
$\displaystyle \bold{a}\cdot \bold{b}=1+1+1+1=4$
Thus,
$\displaystyle 4\leq ||\bold{a}||||\bold{b}||$
Square both sides,
$\displaystyle 16\leq (X+Y+Z+W)\left(\frac{1}{X}+\frac{1}{Y}+\frac{1}{Z} +\frac{1}{W} \right)$
this is also a result of
$\displaystyle (a_1+a_2+\dots+a_n)\left(\frac{1}{a_1}+\frac{1}{a_ 2}+\dots+\frac{1}{a_n}\right)\ge n^2$
assuming that $\displaystyle a_i>0$ for all $\displaystyle i$ such that $\displaystyle 1\le{i}\le{n}$
that can be proved by either using AM-GM or Cauchy