Question 70
Let sequences of real numbers $\left(x_{n}\right)$ and $\left(y_{n}\right)$ satisfy $x_{1}=y_{1}=1$ and
$$
x_{n+1}=\frac{x_{n}+2}{x_{n}+1} \quad \text{ and } \quad y_{n+1}=\frac{y_{n}^{2}+2}{2 y_{n}} \text{ for } n=1,2, \ldots .
$$
Prove that $y_{n+1}=x_{2^{n}}$ holds for $n=0,1,2, \ldots$
$$
x_{n+1}=\frac{x_{n}+2}{x_{n}+1} \quad \text{ and } \quad y_{n+1}=\frac{y_{n}^{2}+2}{2 y_{n}} \text{ for } n=1,2, \ldots .
$$
Prove that $y_{n+1}=x_{2^{n}}$ holds for $n=0,1,2, \ldots$