DE-MO 2005 Problem 6

The sequence $x_0,x_1,x_2,.....$ of real numbers is called with period $p$ ,with $p$ being a natural number, when for each $p\ge2$ , $x_n=x_{n+p}$ . Prove that,for each $p\ge2$ there exists a sequence such that $p$ is its least period and $x_{n+1}=x_n-\frac{1}{x_n}$ $(n=0,1,....)$