Proposition

Suppose $f$ is a nonconstant entire function. Prove that any complex number is arbitrarily close to a number in $f(\mathbb{C})$.

Solution

Let $z_0 \in \mathbb{C}$ and $r > 0$. Suppose that $\abs{f(z) - z_0)} \geq r$ for all $z \in \mathbb{C}$. Then the function $g(z) = \frac{1}{f(z) - z_0}$ is a bounded entire function. By Liouville’s Theorem, $g(z)$ is constant. However, this implies that $f(z)$ is constant, thus this is a contradiction. Therefore, there exists a $z \in \mathbb{C}$ such that $\abs{f(z) - z_0} < r$. In other words, any complex number is arbitrarily closed to a number in $f(\mathbb{C})$.