I'm trying to prove the distrubitivity of multiplication of the ordinals. Here, i have a problem. Does the sup function acts linearly as it does in the other areas of math? It feels correct to write this for example for each fixed $ \alpha $,
$ \sup \{ \alpha \cdot ( \beta \cdot \lambda ) \vert \lambda < \delta \} = \alpha \sup \{ \beta \cdot \lambda ) \vert \lambda < \delta \} $
Am I correct or this is just a bad intiution?
A standard way to prove a property of ordinals is the transfinite induction. The equality given in the question is true almost by definition of multiplication of ordinals (so it turns out I didn't use transfinite induction in my proof). If $\delta=0$ the left and right hand sides are both equal to $0$. If $\delta=\delta'+1$ for some ordinal $\delta'$, both sides are equal to $\alpha(\beta\delta')$ because $\gamma<\delta$ is then equivalent to $\gamma\le\delta'$. Suppose that $\delta$ is a limit ordinal. Then $\beta\delta$ is also a limit ordinal. Note that $$\sup\{\beta\gamma\colon\gamma<\delta\}=\beta\delta$$ and $$\sin\{\alpha\eta\colon\eta<\beta\delta\}=\alpha(\beta\delta)$$ by definition. So both sides are equal to $\alpha(\beta\delta)$.