Ordinal Multiplication is Closed

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

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


 * $x \cdot y \in \operatorname{On}$

Proof
The proof proceeds 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.