Sum of Functions of Exponential Order

Theorem
Let $f, g: \R \to \F$ be functions, where $\F \in \set {\R, \C}$.

Suppose $f$ is of exponential order $a$ and $g$ is of exponential order $b$.

Then $f + g: t \mapsto \map f t + \map g t$ is of exponential order $\max \set {a, b}$.

Proof
Let $t$ be sufficiently large so that both $f$ and $g$ are of exponential order on some shared unbounded closed interval.

By the definition of exponential order: