Complex Exponential is Uniformly Continuous on Half-Planes/Corollary

Corollary to Complex Exponential is Uniformly Continuous on Half-Planes
Let $X$ be a set.

Let $\family {g_n}$ be a family of mappings $g_n : X \to \C$.

Let $g_n$ converge uniformly to $g: X \to \C$.

Let there be a constant $a \in \R$ such that $\map \Re {\map g x} \le a$ for all $x \in X$.

Then $\exp g_n$ converges uniformly to $\exp g$.

Proof
By uniform convergence, there exists $N > 0$ such that $\cmod {\map {g_n} x - \map g x} \le 1$ for all $n > N$.

Then $\map \Re {\map {g_n} x} \le a + 1$.

The result now follows from
 * Complex Exponential is Uniformly Continuous on Half-Planes, applied to the half-plane $\set {z \in \C : \map \Re z \le a + 1}$
 * Uniformly Continuous Function Preserves Uniform Convergence