Is there any way to prove this with only the MVT and IVT? We showed this using the Banach Fixed Point theorem and a cauchy sequence however we were told there is an easier way. Any help would be much appreciated.
Suppose $f$ is differentiable on $(-\infty,\infty)$ and there is a constant $k<1$ such that $|f'(x)|\leq k$ for all real $x$. Show that $f$ has a fixed point.