已知数列 $\left\{x_n\right\}$ 满足 $:x_1=1,x_n=x_{n+1}+\ln \left(1+x_{n+1}\right)\left(n\in {\mathbf{N}}^{*}\right)$.

证明: 当 $n\in {\mathbf{N}}^{*}$ 时,

( I ) $0<x_{n+1}<x_n$;

( II ) $2x_{n+1}-x_n⩽\frac{x_n{x}_{n+1}}{2}$;

( III ) $\frac{1}{2^{n-1}}⩽x_n⩽\frac{1}{2^{n-2}}$.