Let $f$ be an entire function, non constant with the property: The set $\{w : f(w)=0\}$ has ininite elements. Show that for all c there exist a sequence $\{z_n\}$ such that $z_n\to\infty$ and $f(z_n)\to c$.
Anyone could please give me some help for this exercise? Thanks in advance.
Apply the Casoratti-Weierstrass theorem to $g(z)=f\left(\frac1z\right)$. Note that if the Taylor series of $f$ at $0$ is$$a_0+a_1z+a_2z^2+\cdots,$$then infinitely many $a_n$'s are non-zero, because otherwise $f$ would be either the null function (which is constant) or a non-null polynomial (which has only finitely many zeros). But then the Taylor series of $g$ at $0$ is$$a_0+a_1z^{-1}+a_2z^{-2}+\cdots$$ and therefore $g$ has an essential singularity at $0$.