# Thread: proof involving natural numbers

1. ## proof involving natural numbers

How do I prove that N+ (positive natural numbers) are closed under multiplication?

2. Originally Posted by jzellt
How do I prove that N+ (positive natural numbers) are closed under multiplication?

see here. what you want is on the last page. but the proof refers to things on previous pages.

3. Originally Posted by jzellt
How do I prove that N+ (positive natural numbers) are closed under multiplication?

We know there is a unique function $\displaystyle \cdot : \mathbb{N}\times \mathbb{N}\to \mathbb{N}$ such that $\displaystyle \cdot (n,0) = 0\text{ and }\cdot(n,m+1) = \cdot(n,m) + m$ for all $\displaystyle n,m\in \mathbb{N}$. We will write $\displaystyle n\cdot m$ instead of $\displaystyle \cdot (n,m)$. Now we want to prove that if $\displaystyle a,b\in \mathbb{N}^+$ then $\displaystyle a\cdot b\in \mathbb{N}^+$. Well, here is a hint, if $\displaystyle a>0$ then $\displaystyle a=a_0+1$.

### proofs involv

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