Complex Exponential is Uniformly Continuous on Half-Planes

Theorem
Let $a \in \R$.

Then $\exp$ is uniformly continuous on the half-plane $\set {z \in \C : \map \Re z \le a}$.

Proof
Let $\epsilon > 0$.

For $x, y \in \C$ with $\map \Re x, \map \Re y \le a$:

Because Exponential Function is Continuous, there exists $\delta > 0$ such that $\cmod {e^z - 1} < \epsilon$ for $\cmod z < \delta$.

Thus if $\cmod {x - y} < \delta$, $\cmod {e^x - e^y} < e^a \epsilon$.

Thus $\exp$ is uniformly continuous.