Question 46
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 positive integers $n$. Prove that these sequences are convergent and find their limits.
$$
\begin{cases} x_{n+1}y_{n+1} - x_n = 0 \\ x_{n+1}^2 + y_n = 2 \end{cases}
$$
for all positive integers $n$. Prove that these sequences are convergent and find their limits.