VMO 2008 Problem 4

he sequence of real number $(x_n)$ is defined by $x_1 = 0,$ $x_2 = 2$ and $x_{n+2} = 2^{-x_n} + \frac{1}{2}$ $\forall n = 1,2,3 \ldots$ Prove that the sequence has a limit as $n$ approaches $+\infty.$ Determine the limit.