Arithmetic Mean - Geometric Mean Inequality (AM-GM)

The arithmetic mean - geometric mean inequailty is a famous multivariable inequality that allows us to compare the sum of some non-negative numbers with their product, in the presence of some other operations. This amazing multivariable inequality has widespread use across mathematics, including on math contests and olympiads. This proof is even more remarkable, as it uses an exotic technique called Cauchy's forward-backward induction.

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