Cauchy-Schwarz -> AM-HM inequality

[EighthGrader]

Homework Statement

Prove the AM-HM inequality using the Cauchy-Schwarz Inequality.

Homework Equations

Cauchy Schwarz Inequality:

$$\[ \biggl(\sum_{i=1}^{n}a_{i}b_{i}\biggr)^{2}\le\biggl(\sum_{i=1}^{n}a_{i}^{2}\biggr)\biggl(\sum_{i=1}^{n}b_{i}^{2}\biggr)\$$

AM-HM inequality:

$$A(n,a_i) = \frac{a_1 + a_2+\cdots+a_n}{n}\$$


$$H(n,a_i) = \frac{n}{\frac{1}{a_1}+\frac{1}{a_2}+ \cdots+\frac{1}{a_n}}\$$


$$A(k,x_i) \geq H(k,x_i)\$$

The Attempt at a Solution

I just need some tips on how to approach this problem. How do I introduce the term $$n$$ on both sides?

 

[Gregg]

$$\left(\sum _{i=1}^n \frac{1}{a_i}\right)\left(\sum _{i=1}^n a_i\right)\geq \left(\sum _{i=1}^n \frac{\sqrt{a_i}}{\sqrt{a_i}}\right){}^2$$