How do I prove that N+ (positive natural numbers) are closed under multiplication?
Any advice...Thanks in advance.
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 $\displaystyle a=a_0+1$.