# A question on mean value inequality

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?

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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