How would you prove that $$n^{\gamma} = m^{\gamma}$$ for every limit ordinal $\gamma$ and $n,m$ finite ordinals?
It's a rather short solution problem, but I can't construct any slick answer for it. I know very little about ordinal exponentiation, just that $\alpha^{\beta +1} = \alpha^{\beta}\cdot\alpha$ and that if there's a limit ordinal in the exponent we take the $\sup$.
You need to assume that $m,n>1$.
First show that it’s true for $\gamma=\omega$. Then show that if it’s true for some limit $\gamma$, it’s true for $\gamma+\omega$; this is actually a bit like the first step. Finally, show that if it’s true for every limit ordinal less than $\gamma$, and $\gamma$ is a limit of limit ordinals, then it’s true for $\gamma$ as well.