Let x_{1}, x_{2}, \ldots, x_{n} be positive numbers, with n \geq 2. Prove that
\left(x_{1}+\frac{1}{x_{1}}\right)\left(x_{2}+\frac{1}{x_{2}}\right) \cdots\left(x_{n}+\frac{1}{x_{n}}\right) \geq\left(x_{1}+\frac{1}{x_{2}}\right)\left(x_{2}+\frac{1}{x_{3}}\right) \cdots\left(x_{n-1}+\frac{1}{x_{n}}\right)\left(x_{n}+\frac{1}{x_{1}}\right) .