题目

设 $f\left( x\right)$ 在 $\left\lbrack {a,b}\right\rbrack$ 上可导,且 $\left| {{f}^{\prime }\left( x\right) }\right| < 1$ ,当 $x \in \left\lbrack {a,b}\right\rbrack$ 时,有 $a < f\left( x\right) < b,F\left( x\right) = \frac{1}{2}\left\lbrack {x + f\left( x\right) }\right\rbrack$ ,证明:
(I) 存在 ${x}^{ * } \in \left( {a,b}\right)$ ,使得 $F\left( {x}^{ * }\right) = {x}^{ * }$ ;
(II) 对 ${x}_{0} \in \left\lbrack {a,b}\right\rbrack$ ,数列 $\left\{x_{n}\right\}$ 满足 ${x}_{n + 1} = F\left( {x}_{n}\right) \left( {n = 1,2,\cdots }\right)$ ,有 $\mathop{\lim }\limits_{n \rightarrow \infty }{x}_{n} = {x}^{ * }$ .