Exercise 7, § 1

With the notation of exerc. 6 show that if one puts $x_i = a_i / A_n$ the relation $G_n \geq (1-\alpha)A_n$ for $0 \leq \alpha < 1$ implies, for $1 \leq i \leq n$,

Deduce that for each index $i$ one has $1 + x' \leq x_i \leq 1 + x''$, where $x'$ is the root $\leq 0$, and $x''$ the root $\geq 0$, of the equation $(1+x)e^{-x} = (1-\alpha)^n$ (cf. III, p. 115, exerc. 2).