IMC 2011 · Problem 1

Let $(a_n)_{n=0}^{\infty}$ be a sequence with $\frac{1}{2} < a_n < 1$ for all $n \ge 0$. Define the sequence $(x_n)_{n=0}^{\infty}$ by

$$x_0 = a_0, \qquad x_{n+1} = \frac{a_{n+1} + x_n}{1 + a_{n+1}x_n} \quad (n \ge 0).$$

What are the possible values of $\lim\limits_{n \to \infty} x_n$? Can such a sequence diverge?