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 $(f_n)$ be a family of mappings $f_n : X\to\C$.

Let $f_n$ converge uniformly to $f:X\to\C$.

Let there be a constant $a\in\R$ such that $\Re f(x) \leq a$ for all $x\in X$.

Then $\exp f_n$ converges uniformly to $\exp f$.