Cauchy-Schwarz -> AM-HM inequality

  • #1

Homework Statement



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

Homework Equations



Cauchy Schwarz Inequality:

[tex]
\[ \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)\
[/tex]

AM-HM inequality:

[tex]A(n,a_i) = \frac{a_1 + a_2+\cdots+a_n}{n}\[/tex]


[tex]H(n,a_i) = \frac{n}{\frac{1}{a_1}+\frac{1}{a_2}+ \cdots+\frac{1}{a_n}}\[/tex]


[tex]A(k,x_i) \geq H(k,x_i)\[/tex]

The Attempt at a Solution



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

Answers and Replies

  • #2
459
0
[tex] \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 [/tex]
 

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