Function of Exponential Order of Scalar Multiple

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

Let $\lambda$ be a real constant.

Suppose $f(t)$ is of exponential order $a$.

Then the function defined by $t \mapsto f(\lambda t)$ is of exponential order $a\lambda$.

Proof
The result follows by the definition of exponential order.