# "proof of the Cauchy theorem"

Cauchy's Theorem in complex analysis Rudin in proof. Why $g$ is a uniformly continuous function?

• Becasue $h(z_n)\to h(z)$ whenever $z_n\to z$ was shown immediately before stating this. Nov 16, 2014 at 21:46
• Do you know complete proof? For example, why the function g is uniformly continuous.
– reza
Nov 16, 2014 at 22:08

We can prove this immediately. Fix an $\epsilon > 0$. Then for each point $x$ in the compact set, there is an associated $\delta_x$ as in the $\epsilon$-$\delta$ formulation of continuity. Associate to each point $x$ the ball of radius $\delta_x$ centered at $x$. These balls cover the compact set, and therefore a finite subset of them cover the set. Choosing the minimum radius of these covering balls gives the uniform $\delta$ for the $\epsilon$-$\delta$ formulation of uniform continuity.
As $g$ is continuous on a compact set, it is uniformly continuous.