Let $f$ be a holomorphic self-map of $\mathbb{C} \backslash \{ 0 \}, \mathbb{C}$, or the extended complex plane $\overline{\mathbb{C}}$ that is neither injective nor constant. This paper gives new and elementary proofs of the well-known fact that the Julia set of $f$ is a non-empty perfect set and coincides with the closure of the set of repelling cycles of $f$. The proofs use Montel–Caratheodory's theorem but do not use results from Nevanlinna theory.