Application of Casorati–Weierstrass theorem

Question

Let $f$ be an entire function which is not a polynomial. If $B$ is any bounded subset of $\mathbb{C}$, then prove that the set $f(\mathbb{C} \backslash B)$ is dense in $\mathbb C$.

This problem came in isi phd entrance exam once,

(Casorati–Weierstrass) Suppose $f$ is holomorphic in the punctured disc $D_{r}\left(z_{0}\right)-\left\{z_{0}\right\}$ and has an essential singularity at $z_{0}$. Then, the image of $D_{r}\left(z_{0}\right)-\left\{z_{0}\right\}$ under $f$ is dense in the complex plane.

In my problem given that $B$ is any bounded subset of $\mathbb C$, then how can I use the casorati weierstrass theorem in my problem. (Also please tell me that, can this problem be solve without using this theorem)

Answer 1

Since $f$ is entire, there is a power series centered at $0$ such that $f(z)=\sum_{n=0}^\infty a_nz^n$ for each $z\in\Bbb C$. And, since $f$ is not polynomial, $a_n\ne0$ for infinitely many $n$'s. So, if $g(z)=f\left(\frac1z\right)$, $0$ is an essential singularity of $g$, and therefore $g\left(\left\{\frac1z\,\middle|\,z\in\Bbb C\setminus B\right\}\right)$ is dense in $\Bbb C$, by the Casorati-Weierstrass theorem. And I doubt that we can avoid using this theorem in order to prove what you want to prove.