Deriving a relation using the Cauchy Schwarz inequality

I must derive the following relation using the Cauchy Schwarz inequality for any collection of $N$ real numbers $a_1,a_2,...,a_N$:

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

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

I must consider $v \in\mathbb{R}^N$ with components given by the numbers $a_1,a_2,...,a_N$ and find a suitable $w \in\mathbb{R}^N$ such that $v \cdot w=\dfrac{a_1+a_2+\cdots+a_N}{N}$

Some guidance and direction would be greatly appreciated.

• $w:= (1/N, 1/N, \ldots, 1/N)$. Commented Oct 23, 2013 at 13:50
• Ok that works. I'm using $|v \cdot w|^2 ≤ ||v||^2 ||w||^2$ and so acquiring the LHS is obvious. What would I need to do to derive the relation? Commented Oct 23, 2013 at 13:55
• Ignore that. I now have $v \cdot w$, $v \cdot v$ and $w \cdot w$. Commented Oct 23, 2013 at 14:06

Hint: proceed with the vectors $(a_1,a_2,\cdots,a_n)$ and $(1,1,\cdots,1)$.