Exercise 14, § 2

Let $\Gamma$ be an equicontinuous group of homeomorphisms of $\mathbf{R}$. Show that if a homeomorphism $u \in \Gamma$ leaves at least one point of $\mathbf{R}$ fixed, and if $u$ is increasing, then $u$ is the identity mapping (show that otherwise the group generated by $u$ is not equicontinuous at a suitably chosen fixed point of $u$). If $u$ is decreasing, then $u^2$ is the identity mapping.