# Epsilon number cannot be a successor?

Let $\omega$ be the canonical representation of the natural numbers. A epsilon number $\alpha$ is a ordinal number such that $\alpha=\omega^\alpha$ holds. I suspect that a epsilon number cannot be a successor, but I am unable to prove it. Could anyone help me?

HINT: Show that $\omega^\alpha$ is a limit ordinal by induction on $\alpha$. In particular $\varepsilon$ numbers are limit ordinals.