Let $a,b$ be two $n$ dimensional vectors, we want to show that
$(\sum\limits_k a_k b_k)^2 \leq \sum\limits_k b_k a_k^2 \sum\limits_k b_k$
Recall the Cauchy Schwarz inequality is given as
$|\langle x,y \rangle| \leq \|x\|\|y\|$
then let $x = a,y = b$
Then
$|\langle a,b \rangle| \leq \|a\|\|b\|$
= $|a^Tb| \leq \sqrt{a^Ta} \sqrt{b^Tb}$
= $|a^Tb|^2 \leq a^Ta b^Tb$
$\Rightarrow$ $|\sum\limits_k a_k b_k|^2 \leq \sum\limits_k a_k^2 \sum\limits_k b_k^2$
At this point how can we eliminate the absolute value sign for the first term and massage the right hand side to what we wish to show?