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I need to prove the AM-GM Inequality using a few specific lemmas that I have already proven. I'm mostly just unsure what to do next and how to tie it all together at the end to finish the proof.

Here are the lemmas that I have already proven.

  1. The theorem is true for $n=2$.
  2. Suppose the statement is true for $n=k$. Then it is true for $n=2k$.

  3. Suppose $m<n$ and $x_{1} ,...,x_{m}>0$. Then the geometric mean of n numbers is $g = \sqrt[m]{x_{1}*...*x_{m}}$.

  4. Suppose $m<n$ and the theorem is true for any $n$ numbers. Then the theorem is true for any $m$ numbers.

Any help would be greatly appreciated!

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Use induction with lemma 1 as a base case and lemma 2 to drive the induction step, proving AM-GM for any $n$ equal to a power of 2.

For any $m$ not a power of 2, AM-GM holds for some $n$ a power of 2 greater than $m$. By lemma 4, AM-GM holds for $m$ numbers.

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  • $\begingroup$ So I wouldn't necessarily need to use anymore principles or properties? The four lemmas I have already proven should be enough with your added explanation at the end? Sorry if I am misinterpreting! $\endgroup$ – ej313 Jun 15 '14 at 3:40
  • $\begingroup$ @ej313: You've proven pretty much everything you need. $\endgroup$ – user2357112 Jun 15 '14 at 3:43

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