Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a function such that

$$f\left(z+\omega_{1}\right)=f(z), \quad f\left(z+\omega_{2}\right)=f(z)$$

where $\omega_{1}, \omega_{2} \in \mathbb{C} \backslash\{0\}$ and $\omega_{1} / \omega_{2}$ is not real. Show that if $f$ is analytic on $\mathbb{C}$ then it is a constant. [Liouville's theorem may be used if stated.] Give an example of a non-constant meromorphic function which satisfies (1).