I am trying to prove the Euclidean algorithm for ordinals. More specifically: For any ordinals $\alpha, \beta$ where $\beta >0$, there are unique $\gamma, \delta$ such that $\alpha = \beta\gamma+\delta$ and $\delta<\beta$.
For now, I have difficulties. For example, I am not sure what to do when, $\alpha$ is a limit ordinal and $\beta$ is finite (a natural number).. since "diminishing" multiples of $\beta$ from $\alpha$ will not help.. and I don't really have an action of division...
Can anyone help? maybe give me a clue or a direction??