Complex Exponential is Uniformly Continuous on Half-Planes

Theorem
Let $a\in\R$.

Then $\exp$ is uniformly continuous on the half-plane $\{z\in\C : \Re(z) \leq a\}$.

Proof
Let $\epsilon>0$.

For $x,y\in\C$ with $\Re (x),\Re (y)\leq a$,

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

Thus if $|x-y|<\delta$, $\vert e^x-e^y\vert < e^a\epsilon$.

Thus $\exp$ is uniformly continuous.