VMO 2009 Problem 2

The sequence $\{x_n\}$ is defined by $$\left\{ \begin{array}{l}x_1 = \frac{1}{2} x_n = \frac{{\sqrt {x_{n - 1} ^2 + 4x_{n - 1} } + x_{n - 1} }}{2} \end{array} \right.$$ Prove that the sequence $\{y_n\}$ , where $y_n=\sum_{i=1}^{n}\frac{1}{{{x}_{i}}^{2}}$ , has a finite limit and find that limit.