<s>[INST] Proofs of the Cauchy-Schwarz Inequality? How many proofs of the Cauchy-Schwarz inequality are there? Is there some kind of reference that lists all of these proofs?
 [/INST] Yet another proof can be done by AM-GM inequality---
In all that follows,all summations run from $i=1$ to $n$.
Let $X^2=\sum{x_i}^2 , Y^2=\sum{y_i}^2\tag{1}$
By Arithmetic-mean-Geometric-mean(AM-GM) inequality(i.e. AM$\ge$GM ),we have
$\frac{a+b}{2}\ge\sqrt{ab}\tag{2}$
Putting $a=x_i^2/X^2$ and $b=y_i^2/Y^2$ in $(2)$,we get
$$\frac{(x_i^2/X^2+y_i^2/Y^2)}{2}\ge \frac{x_iy_i}{XY}$$
Taking summation on both sides, we obtain 
$$\sum\frac{(x_i^2/X^2+y_i^2/Y^2)}{2}\ge \sum\frac{x_iy_i}{XY}\tag{3}$$
Since X and Y are already summed over all $i$s,they are independent of the individual $x_i$s and $y_i$s and can be taken outside the summations.Therefore,the quantity on LHS can be written as,
$$\frac{\sum x_i^2}{2X^2}+\frac{\sum y_i^2}{2Y^2}=\frac{X^2}{2X^2}+\frac{Y^2}{2Y^2}=1/2+1/2=1$$ where the first equality is a consequence of $(1)$.
$\therefore$ LHS of (3) is 1. Hence,$(3)$becomes,
$$1 \ge \sum\frac{x_iy_i}{XY}$$
Arranging the inequality properly and putting in X and Y from $(1)$ and finally squaring both sides,we get
$$\left( \sum_i x_i y_i \right)^2 \le \left( \sum_i x_i^2 \right) \left( \sum_i y_i^2 \right) 
$$
which is the Cauchy-Schwarz inequality.
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