# Fixed Point problem of walter Rudin

Suppose $f$ is differentiable on $(-\infty , \infty)$. We say that $x$ is a fixed point if $f(x) = x$. If there is a constant $A$ such that $0 <A <1$ and $|f'(x)| \leq A$ for all real $x$. Then there is a fixed point $x$ such that $x = \lim_{n\to \infty} x_n$ , where $x_1$ is any arbitrary real number and $x_{n+1} = f(x_n)$ for all $n= 1 , 2, \dots$

• But my doubt is that if $f(x) = x + (1 + e^x)^{-1}$ and $0 <f'(x) <1$ and $f(x)$ has no fixed point .Both statement are contradict here.

Please clear my doubt. Thank you