Proof of AM-GM by Induction
by djmathman, Aug 7, 2024, 7:57 PM
and not Cauchy Induction. This proof can be found in Chapter 27 of the kind of insane book Classical and New Inequalities in Analysis.
Let
be the maximum value of 
subject to the conditions 
and 
. We will crucially use the fact that 
is homogeneous, i.e. 
.
Assume by scaling that
. Fix ![$x_n = t\in[0,1]$](//latex.artofproblemsolving.com/f/2/f/f2f2866e4c4e81ebca79ca5baf4f554379466b3a.png)
. Then 
, and so the product is at least
![\[
t f_{n-1}(1 - t) = t(1 - t)^{n-1} f_{n-1}(1).
\]](//latex.artofproblemsolving.com/0/1/d/01da829e8c351ed062798f8772beac46e3dfb187.png)
The minimum value of 
in the interval ![$t\in[0,1]$](//latex.artofproblemsolving.com/e/7/0/e70754c1efb07bd54438989e7e001bafb98b429a.png)
is 
, achieved when 
. (This is easiest to prove by differentiating the analogous expression 
.) Therefore
![\[
f_n(1) = \max_{t\in[0,1]}t(1-t)^{n-1}f_{n-1}(1) = \frac{(n-1)^{n-1}}{n^n}f_{n-1}(1).
\]](//latex.artofproblemsolving.com/b/2/9/b290d96bbaaa5808dc3b79eddeacfbb4ae2f2f29.png)
A quick induction argument implies 
, which is equivalent to AM-GM.
Let
be the maximum value of 
subject to the conditions 
and 
. We will crucially use the fact that 
is homogeneous, i.e. 
.Assume by scaling that
. Fix ![$x_n = t\in[0,1]$](http://latex.artofproblemsolving.com/f/2/f/f2f2866e4c4e81ebca79ca5baf4f554379466b3a.png)
. Then 
, and so the product is at least![\[
t f_{n-1}(1 - t) = t(1 - t)^{n-1} f_{n-1}(1).
\]](http://latex.artofproblemsolving.com/0/1/d/01da829e8c351ed062798f8772beac46e3dfb187.png)
The minimum value of 
in the interval ![$t\in[0,1]$](http://latex.artofproblemsolving.com/e/7/0/e70754c1efb07bd54438989e7e001bafb98b429a.png)
is 
, achieved when 
. (This is easiest to prove by differentiating the analogous expression 
.) Therefore![\[
f_n(1) = \max_{t\in[0,1]}t(1-t)^{n-1}f_{n-1}(1) = \frac{(n-1)^{n-1}}{n^n}f_{n-1}(1).
\]](http://latex.artofproblemsolving.com/b/2/9/b290d96bbaaa5808dc3b79eddeacfbb4ae2f2f29.png)
A quick induction argument implies 
, which is equivalent to AM-GM.This post has been edited 1 time. Last edited by djmathman, Aug 7, 2024, 7:57 PM
April 2025
November 2024