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Homework Help: Cauchy-Schwarz -> AM-HM inequality

  1. Jun 27, 2010 #1
    1. The problem statement, all variables and given/known data

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

    2. Relevant 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:

    [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]

    3. 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?
  2. jcsd
  3. Jun 27, 2010 #2
    [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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