Suppose f is Complex differentiable everywhere and that Lim|z|→∞}f(z)=1
Prove f(z) is constant
First proof: Liouville's theorem states that a holomorphic (analytic) function which is bounded must be constant, and that's exactly what happens with our function: in some circle
Second proof: By Cauchy's integral representation and by the estimation lemma: , so constant.
Tonio