This is a problem from Spivak's Calculus.
If $a_1, \dots, a_n \ge 0$, then the "arithmetic mean"
$$ A_n= \frac{a_1+\cdots +a_n}{n}$$
and "geometric mean"
$$ G_n= \sqrt[n]{a_1 \dots a_n}$$
satisfy $G_n \le A_n$.
Suppose that $a_1 \lt A_n$. Then some $a_i \gt A_n$; for convenience, say $a_2 \gt A_n$. Let $\bar a_1=A_n$ and let $\bar a_2=a_1+a_2- \bar a_1$. Show that $\bar a_1 \bar a_2 \ge a_1a_2$.(This part is easily done). Why does repeating this process enough times eventually prove that $G_n \le A_n$? (This is another place where it is a good exercise to provide a formal proof by induction, as well as an informal reason.) When does equality hold in the formula $G_n \le A_n$?
I don't see how I can generalize this process to achieve the result. I would appreciate any hint, suggestions, or solutions.
