Complex Exponential is Uniformly Continuous on Half-Planes/Corollary

Lemma to Uniform Absolute Convergence of Infinite Product of Complex Functions
Let $X$ be a set.

Let $(g_n)$ 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 $\Re( g(x)) \leq a$ for all $x\in X$.

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

Proof
By uniform convergence, there exists $N>0$ such that $|g_n(x)-g(x)|\leq1$ for all $n>N$.

Then $\Re(g_n(x))\leq a+1$.

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