A little confusion about AM-GM proof

Question

In Cauchy's forward-backward induction proof, why can we substitute $x_k=\frac{x_1+x_2+ ... +x_{k-1}}{k-1}$ without losing generality?

Answer 1

We can not substitute it of course in the general case, but if we'll prove that for $P(k): \frac{x_1+x_2+...+x_k}{k}\geq\sqrt[k]{x_1x_2...x_k}$ and fo $x_i>0$ the following statements are true: $$P(2)$$ $$P(k)\Rightarrow P(2k)$$ and $$P(k)\Rightarrow P(k-1)$$ then it's obvious that we'll prove AM-GM.

For the proof of the last statement we can assume $x_k=\frac{x_1+x_2+...+x_{k-1}}{k-1}$ because we assumed that $P(k)$ is true for all positive $k$ numbers $x_i$ and from here

it's true for $k$ positive numbers $x_1$,...,$x_{k-1}$,$\frac{x_1+x_2+...+x_{k-1}}{k-1}$, which gives that $P(k-1)$ is true.