Exercise 23, § 2

Let K be a maximal ordered field and let $a_1, \ldots, a_n, b_1, \ldots, b_n$ be 2n elements of K such that $\sum a_i \neq 0$ and $b_1 < b_2 < \ldots < b_n$. Show that for any integer $m \geq 1$ the number $v$ of roots in K, counted with multiplicity, of the polynomial

is at most equal to the number of variations w of the sequence

and that the difference $w - v$ is even. (Argue by induction on $n$, applying Ex. 13 (VI, p. 40) to a rational function of the form $f(T)/(T - c)^m$.)