RO-NMO 2000 Problem 112

Study the convergence of a sequence $\left( x_n\right)_{n\ge 0}$ for which $x_0\in\mathbb{R}\setminus\mathbb{Q} ,$ and $x_{n+1}\in \left\{ \frac{x_n+1}{x_n} , \frac{x_n+2}{2x_n-1}\right\} ,$ for all $n\ge 1.$