USAMO 1993 Problem 5

A sequence $\mathrm{x}_{\mathrm{n}}$ of positive reals satisfies $\mathrm{x}_{n-1} \mathrm{x}_{\mathrm{n}+1} \leq \mathrm{x}_{\mathrm{n}}{ }^{2}$. Let $\mathrm{a}_{\mathrm{n}}$ be the average of the terms $\mathrm{x}_{0}, \mathrm{x}_{1}, \ldots, \mathrm{x}_{\mathrm{n}}$ and $b_{n}$ be the average of the terms $x_{1}, x_{2}, \ldots, x_{n}$. Show that $a_{n} b_{n-1} \geq a_{n-1} b_{n}$.