VMO 2014 Problem 1

Let $({{x}_{n}}),({{y}_{n}})$ be two positive sequences defined by ${{x}_{1}}=1,{{y}_{1}}=\sqrt{3}$ and $$\begin{cases} {{x}_{n+1}}{{y}_{n+1}}-{{x}_{n}}=0 x_{n+1}^{2}+{{y}_{n}}=2 \end{cases}$$ for all $n=1,2,3,\ldots$ . Prove that they are converges and find their limits.