$\text{span} \{ e_t \cdot w : t \in \mathbb{R} \}$ dense in $C_0(\mathbb{R}_+)$.

Question

Let $\mathbb{R}_+ := [0,\infty )$ and let $w \in C_0(\mathbb{R}_+)$ be any function with $w(x) \neq 0$ for all $x \geq 0$.

Why is $\text{span} \{ e_t \cdot w : t \in \mathbb{R} \}$, where $e_t(x) := \exp^{i tx}$ dense in $C_0(\mathbb{R}_+)$.