Let $\alpha, \gamma$ be real numbers such that $0<\alpha<1$ and $\gamma>0$. Consider the sequence of real numbers given by $$ \begin{cases} x_0\ne 0&\\ x_{k+1}=x_k\left(1-\frac{\gamma(1+\alpha)}{|x_k|^{1-\alpha}}\right) \quad (k\in \mathbb{N}).& \end{cases} $$ Suppose that $x_k\ne 0$ for all $k\in \mathbb{N}$. Prove that :
The sequence $\{x_k\}_{k\in\mathbb{N}}$ does not converge.
The sequence $\{|x_k|\}_{k\in\mathbb{N}}$ converges to $[(1/2)(1+\alpha)\gamma]^{1/(1-\alpha)}.$