A sequence of finite recursive sequences [closed]

For a positive integer $n$ define a sequence $x_0=\frac{1}{2}$ and for $1 \leq k \leq n$ define $$x_k=x_{k-1}+\frac{x_{k-1}^2}{n}$$ Show that $x_n$ lies in the open interval $\left(1-\frac{1}{n}, 1\right)$.