Originally Posted by

**MichaelH** So I am trying to derive the following relation using Cauchy Schwarz inequality for any collection of N real numbers:

$\displaystyle \left(\frac{a_1+a_2+\cdots+a_N}{N}\right)^2 =< \frac{a^2_1+a^2_2+\cdots+a^2_N}{N}.$

This says that the square of the average is less or equal than the average of the squares.

So I want to use $\displaystyle |v \cdot w|^2 =(v \cdot v)*(w \cdot w)$ to derive this.

So $\displaystyle v \cdot w$ needs to be $\displaystyle \frac{a_1+a_2+\cdots+a_N}{N}$ therefore, I have $\displaystyle v \cdot v$ as $\displaystyle a^2_1+a^2_2+\cdots+a^2_N$ and $\displaystyle w \cdot w$ as $\displaystyle \frac{1}{N^2}$ providing that:

$\displaystyle v: a_1+a_2+\cdots+a_N$

$\displaystyle w: \frac{1}{N}$

Am I going in the right direction? Can anyone provide me some guidance?