Let $f$ be entire and non-constant. For any positive real number $c$, show that the closure of the set $\{z\mid |f(z)|<c\}$ is the set $\{z\mid |f(z)|\leq c\}$.
This is an exercise question in the Conway's functions of one complex variable 1. I am supposed to use the maximum modulus principle to solve this problem. This problem looks intuitively clear but I cannot find a way to rigorously prove it. Could anyone help me with it..
Hint: the fact that $|f|$ does not have a local minimum at a point where $|f(z)|=c$ means there are $w$ near $z$ with $|f(w)| < c$.