Scalar Multiple of Function of Exponential Order

Theorem
Let $f: \R \to \mathbb{F}$ be a function, where $\mathbb{F} \in \left \{{\R,\C}\right\}$.

Let $\lambda$ be a complex constant.

Suppose $f$ is of exponential order $a$.

Then $\lambda f$ is also of exponential order $a$.

Proof
If $\lambda = 0$, the theorem holds trivially.

Let $\lambda \ne 0$.