Product of Functions of Exponential Order

Theorem
Let $f,g \left({t}\right): \R \to \mathbb{F}$ be functions, where $\mathbb{F} \in \left \{{\R,\C}\right\}$.

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

Then $fg: t \mapsto f\left({t}\right)g\left({t}\right)$ 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: