# Finite Ordinal Times Ordinal/Lemma

## Lemma

Let $m$ be a finite ordinal.

Let $m \ne 0$, where $0$ is the zero ordinal.

Then:

$m \times \omega = \omega$

where $\omega$ denotes the minimal infinite successor set.

## Proof

 $\displaystyle \forall n \in \omega \ \$ $\displaystyle m \times n$ $\in$ $\displaystyle \omega$ Natural Number Multiplication is Closed $\displaystyle \bigcup_{n \mathop \in \omega} \left({ m \times n }\right)$ $\le$ $\displaystyle \omega$ Supremum Inequality for Ordinals $\displaystyle \implies \ \$ $\displaystyle m \times \omega$ $\le$ $\displaystyle \omega$ Definition of Ordinal Multiplication

Also, $\omega \le \left({ m \times \omega }\right)$ by Subset is Right Compatible with Ordinal Multiplication.

The lemma follows from the definition of equality.

$\Box$