Natural Number Multiplication is Closed

Theorem
Let $m$ and $n$ be natural numbers.

Let $\omega$ denote the minimal infinite successor set.


 * $m \times n \in \omega$

where $\times$ denotes natural number multiplication.

Proof
Let $g_m: \omega \to \omega$ be the mapping as specified in the definition of multiplication.

Then in particular, $g_m \left({n}\right) \in \omega$.

Hence, by definition of $\times$:


 * $m \times n = g_m \left({n}\right) \in \omega$