Sum/Product of two natural numbers is a natural number I wanted to prove that the sum and the product of two natural numbers is a natural number. Intuitively it's clear to my why that is true, however I couldn't prove it.
So our lecturer first defined what an inductive set is. Then he defined the Natural numbers as the intersection of all inductive sets. Then we proved the induction principle and $\forall n \in \mathbb{N} , 1\leq n$ .
So this is the information we have so far. But I am having difficulty proving that the sum/product of two natural numbers is a natural number. Can someone give me a clue how to prove this?
I am trying to prove this with the definition of a Natural numbers, but it's not working so far...
Thank you.

How are sum/product operations in $\mathbb{N}$ defined in your context?
Let $n,m\in\mathbb{N}$.
Write $s(n)=n\cup \{n\}$. Observe $s(n)\in\mathbb{N}$ by definition of inductive set, and recall
$$m\neq 0 \implies \ \exists l\in\mathbb{N} \big(m=s(l)\big).$$
Both addition and multiplication are binary operations in $\mathbb{N}$ defined recursively  as follows:
Addition (or "sum"):


*

*If $m=0$, then $n+ m := m$

*If $m\neq 0$, then $n+ m := s(n+l)$
Multiplication (or "product"):


*

*If $m=0$, then $n\cdot m = 0$

*If $m\neq 0$, then $n\cdot m = n\cdot l + n$
Thus, by this definition and using induction, the result of both sum and product are also natural numbers.
