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It is known that mean value inequality is very useful. It is:

For any $0 \le a_i (i=1,2,\dots,n)$, $$ a_1 a_2\dots a_n\le (\frac{a_1+a_2+\dots + a_n}{n})^n \tag1 $$

My question is: how many ways by which the mean value inequality can be proved?

Thanks for your help.

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@Paul - set $\lambda_i = 1/n$ in the proofs that SYZ's comment links to in order to make the connection – Hans Engler May 31 '13 at 1:08
The inequality don't need to prove. The proof strategy is enough for me. Of course, more details proof are welcome. – Paul May 31 '13 at 1:12
@Paul I saw the tag Integral-inequality, and though I might have misunderstood you post. But I do believe that you are referring to the Arithmetic Mean-Geometric Mean inequality, in which case you can still refer to this AopsWiki page, scroll down and the section below details 2 proofs of non-weighted AM-GM (the one you detailed above). – A Nonny May 31 '13 at 1:17
Wikipedia article gives several proofs of AM-GM nequality. – Martin Sleziak May 31 '13 at 3:52
up vote 2 down vote accepted

One method, which is often employed to prove AM-GM, is Cauchy induction. Such proof is sketched in this answer, this answer, this answer, Wikipedia and in many other places.

More about Cauchy induction:

  • Cauchy induction at AoPS
  • Proof of AM-GM using this type of induction at Wikipedia. They call this technique forward-backward-induction.
  • One section of Pete L. Clark's notes on induction is devoted to this type of induction. He calls it upward-downward induction.
  • Perhaps also this question
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