Product of Functions of Exponential Order

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

Let $f$ be of exponential order $a$ and $g$ be of exponential order $b$.

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

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: