Exercise 8, § 1

Consider a sequence of sets $D_k$ ($k \geq 1$) and for each $k$ consider two functions $(x_1, \ldots, x_k) \mapsto f_k(x_1, \ldots, x_k),\ (x_1, \ldots, x_k) \mapsto g_k(x_1, \ldots, x_k)$ with real values, where $(x_1, \ldots, x_k) \in D_1 \times D_2 \times \cdots \times D_k$. Suppose that for every $k$ there is a real function $F_k$ defined on $\mathbf{R}$ such that for every $\mu \in \mathbf{R}$ and arbitrary $a_1 \in D_1, \ldots, a_{k-1} \in D_{k-1}$ one has

(By convention one takes $f_0 = 1$.) Show if the sequences of real numbers $(\mu_k)$ and $(\delta_k)$ satisfy the conditions $F_1(\mu_1) = 0$ and $F_k(\mu_k) = \delta_k$ then one has the inequalities for all values of $n$ and of $x_k \in D_k$.