TITLE: Distributivity of ordinal arithmetic
QUESTION [8 upvotes]: Let greek letters be ordinals. I want to prove $\alpha(\beta + \gamma) = \alpha\beta + \alpha\gamma$ by induction on $\gamma$ and I already know it holds true for $\gamma = \emptyset$ and $\gamma$ a successor ordinal. Let $\gamma$ be a limit ordinal. I found
$$
\alpha(\beta + \gamma) = \alpha \cdot \sup_{\epsilon < \gamma} (\beta + \epsilon) = \sup_{\epsilon < \gamma} (\alpha(\beta + \epsilon)) = \sup_{\epsilon < \gamma} (\alpha\beta + \alpha\epsilon) = \alpha\beta + \alpha\gamma,
$$
but I am suddenly doubting if the second equality is justified.
Question: Is the second equality correct?

REPLY [7 votes]: Yes, it is correct. It may be more obvious, if we arrange things a bit differently: If $\gamma$ is a limit ordinal, then $\beta+\gamma$ is a limit ordinal as well. So by definition of ordinal multiplication we have 
$$
\begin{align*}
\alpha \cdot (\beta + \gamma) &= \sup_{\delta < \beta + \gamma} \alpha \cdot \delta \\
&= \sup_{\epsilon < \gamma} \alpha \cdot (\beta + \epsilon).
\end{align*}
$$