I'm a bit stuck on telling what the ordinal $\epsilon_0 \cdot \omega$ is.
$$\epsilon_0 = \sup \{1, \omega, \omega^{\omega}, \omega^{\omega^{\omega}}, \dots \}$$
so
$$\epsilon_0 \omega = \sup \{\omega, \omega^2, \omega^{\omega + 1}, \omega^{\omega^{\omega} + 1}, \dots \}$$
My confusion is whether or not in the limit these two sequences end up resulting in the same ordinal, i.e. is $\epsilon_0 \cdot \omega = \epsilon_0$?