A question on mean value inequality

Question

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.

Answer 1

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