An elementary proof for a generalization of a Pohst's inequality

Let $P_n(y_1,\ldots,y_n):= \prod_{1\leq i<j\leq n}\left( 1 -\frac{y_i}{y_j}\right)$ and $P_n:= \sup_{(y_1,\ldots,y_n)}P_n(y_1,\ldots,y_n)$ where the supremum is taken over the $n$-ples $(y_1,\ldots,y_n)$ of real numbers satisfying $0 <|y_1| < |y_2|< \cdots < |y_n|$. We prove that $P_n \leq 2^{\lfloor n/2\rfloor}$ for every $n$, i.e., we extend to all $n$ the bound that Pohst proved for $n\leq 11$. As a consequence, the bound for the absolute discriminant of a totally real field in terms of its regulator is now proved for every degree of the field.