Question 81
A sequence $\{x_n\mid n=0,1,2,\ldots\}$ of real numbers is defined in terms of a real parameter $a$ by
$$\begin{cases}x_0=a,\\x_{n+1}=2-x_n^2\quad\text{for }n\geq1.\end{cases}$$
a. Find all $a$ for which $x_n$ is constant, that is, $x_n=a$ for every $n$.
b. Prove that for one of the values found in (a), called $y$, if $|a|\lt |y|$ then $|x_n|\lt |y|$ for every $n$.
c. Prove that if $|a|\gt |y|$, then $x_n$ is strictly decreasing.
$$\begin{cases}x_0=a,\\x_{n+1}=2-x_n^2\quad\text{for }n\geq1.\end{cases}$$
a. Find all $a$ for which $x_n$ is constant, that is, $x_n=a$ for every $n$.
b. Prove that for one of the values found in (a), called $y$, if $|a|\lt |y|$ then $|x_n|\lt |y|$ for every $n$.
c. Prove that if $|a|\gt |y|$, then $x_n$ is strictly decreasing.