On proving $\omega^{\epsilon} = \epsilon$

Question

How can I prove that $$\omega^{\epsilon} = \epsilon$$ where $\omega =\{0, 1, 2, 3, 4, ...\}$ and $\epsilon = \{\omega, \omega^{\omega}, \omega^{\omega^{\omega}}, \omega^{\omega^{\omega^\omega}}, ...\}$

Answer 1

As $\varepsilon_0 = \sup \{\omega, \omega^\omega, \omega^{(\omega^\omega)}, \dots \}$ we know (by the recursive definition of ordinal exponentiation) that $\omega^{\varepsilon_0} = \sup \{\omega^\omega, \omega^{(\omega^\omega)}, \dots,\}$, which is just the supremum of the tail of the same set, so that it also equals $\varepsilon_0$.