Hi, indeed i was trying to solve that problem from Eccles' book. And i must say, i find both proofs (the one from Cauchy and the one from tonio given here) quite difficult to follow.

But i guess these kind of problems are the hardest ones in the book, it is not really necessary to solve them to understand the basic concepts, but it is still fun to try

tonio: i tried to follow your reasoning but i wasnt able to completely follow it

i thought that if

$\displaystyle

y_i=\frac{x_i}{\left(\prod\limits_{k=1}^nx_k\right )^{1\slash n}}

$

then

$\displaystyle

\sum\limits_{k=1}^ny_k=\frac{\sum\limits_{k=1}^nx_ k}{\left(\prod\limits_{k=1}^nx_k\right)^{1\slash n}}

$

$\displaystyle

\frac{\sum\limits_{k=1}^nx_k}{\left(\prod\limits_{ k=1}^nx_k\right)^{1\slash n}}\geq n \geq 1

$

$\displaystyle

\frac{\frac{1}{n}\sum\limits_{k=1}^nx_k}{\left(\pr od\limits_{k=1}^nx_k\right)^{1\slash n}} \geq 1

$

therefore

$\displaystyle \frac{1}{n}\sum\limits_{k=1}^nx_k \geq {\left(\prod\limits_{k=1}^nx_k\right)^{1\slash n}}\$

i dont know if this is correct, but i could not figure out this step:

$\displaystyle

x_2+...+x_{n-1}+x_1x_n\geq n-1$

santiagos11: did you manage to solve the last problem of chapter 1?