Show that ordinal addition is associative

Question

I want to show that $(\alpha+\beta)+\gamma= \alpha+(\beta+\gamma)$.

I'm carrying out the proof using transfinite induction, I've been successful for the first two steps i.e- For the 0 case and the successor case. I'm struggling to prove associativity for the limit ordinal case.

Any help will be much appreciated, Thanks

Answer 1

Hint: Proceed by induction on $\gamma$. Prove that for limit ordinals $\gamma$, $\beta + \gamma$ is always a limit ordinal. Now observe that, by our induction hypothesis,

$$ \begin{align*} (\alpha + \beta) + \gamma & = \sup \{ (\alpha + \beta) + \delta \mid \delta < \gamma \} \\ &= \sup \{ \alpha + (\beta + \delta) \mid \delta < \gamma \} \\ &= \alpha + \sup \{ \beta + \delta \mid \delta < \gamma \} \\ &= \alpha + (\beta + \gamma). \end{align*} $$