$AM$- $QM$ inequality

Question

Show that $AM \leq QM$ is a direct consequence of the Cauchy- Schwarz inequality.

I have been trying this one but I couldn't solve. Please help.

Answer 1

The Cauchy-Schwartz inequality is :$$\sum_{i=1}^n a_ib_i \leq \Bigg(\sum_{i=1}^n a_i^2\Bigg)^{\frac{1}{2}} \Bigg(\sum_{i=1}^n b_i^2\Bigg)^{\frac{1}{2}}$$

Take $b_i=1, \forall i=1,2,\cdots,n$.

Then $$a_1+a_2+\cdots+a_n=\sum_{i=1}^n a_i \leq \Bigg(\sum_{i=1}^n a_i^2\Bigg)^{\frac{1}{2}} \Bigg(\sum_{i=1}^n1\Bigg)^{\frac{1}{2}}=\sqrt{a_1^2+\cdots+a_n^2}.\sqrt{n}$$

Divide by $n$ to see $$\frac{a_1+a_2+\cdots+a_n}{n} \leq \sqrt{\frac{a_1^2+\cdots+a_n^2}{n}}$$