Exercise 6, § 1

Denote by $A_n$ and $G_n$ the usual arithmetic and geometric means of the first $n$ numbers of a sequence $(a_k)$ ($k \geq 1$) of positive numbers. Show that

with equality only if $a_{n+1} = G_n$ (put $a_{n+1} = x^{n+1},\ G_n = y^{n+1}$).