Ordinal Multiplication is Closed

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

Let $\On$ denote the class of all ordinals.


 * $x \cdot y \in \On$

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.