Question 1
Let positive sequences $(x_n),(y_n)$ be determined by $x_1=1$, $y_1=\sqrt3$ and
$$\begin{cases}x_{n+1}y_{n+1}-x_n=0,\\x_{n+1}^2+y_n=2\end{cases}\qquad(n=1,2,\ldots).$$
Prove that both sequences converge and find their limits.
$$\begin{cases}x_{n+1}y_{n+1}-x_n=0,\\x_{n+1}^2+y_n=2\end{cases}\qquad(n=1,2,\ldots).$$
Prove that both sequences converge and find their limits.