Ordinal Multiplication is Closed

Theorem
Let $x$ and $y$ be ordinals.

Let $\operatorname{On}$ denote the ordinal class.


 * $\displaystyle ( x \cdot y ) \in \operatorname{On}$

Proof
By Transfinite Induction on $y$.

Basis for the Induction
This proves the basis for the induction.

Induction Step
This proves the induction step.

Limit Case
This proves the limit case.