What's the proof that the sum and multiplication of natural numbers is a natural number? I'm explaining the construction of the natural numbers to someone and I'm asking him to show where to find $C$ and $F$ with $a,b,g,h\in \mathbb{N}$ in:
$$a+b=C$$
$$g*h=F$$
I know intuitively (and from some readings) that $C$ and $F$ are natural numbers too. But I don't know what is the proof for it, I'm thinking that every sum or multiplication of natural numbers is a natural number, I just don't know why.
Until now I could only elaborate a geometric demonstration that seems poor to me, see:

The result of the sum of two natural numbers seems certainly to be a natural number: 
$$a+b=\begin{matrix}
{*}&{*}&{...}&{+}&{*}&{*}&{...}\end{matrix}\tag{1.0}$$
For multiplication:
$$a \times b=\begin{matrix}
{*}&{*}&{\cdots}\\ 
{*}&{*}&{\cdots}\\
{\vdots}&{\vdots}&{\ddots}\\
\end{matrix}$$
Then moving each horizontal line to form a new line of height one:
$$a\times b=(* \;\; *\;\; \cdots)+(* \;\; *\;\; \cdots)+(\;\;\vdots \;\;\; \vdots\;\; \ddots)$$
We can use the property given in $(1.0)$ to show the desired result.

The problem for me is that I think I'm pushing $(1.0)$ through intuition and I don't feel the legitimacy of it. 
 A: The first part is a comment rather than an answer, but I don't have enough points to comment.
It is impossible to answer this question mathematically without more information about what your definition of "natural number", "sum" and "multiplication" are. It is closely tied to the question of how you choose to present the foundations of mathematics.
The most straightforward way to define the sum of two numbers $m$ and $n$ is to take two disjoint sets $A$ and $B$ (disjoint means with no common elements), with $m$ and $n$ elements respectively, and to define $m + n$ as the number of elements in the set $A \cup B$. The product $mn$ is the number of elements in the Cartesian product $A \times B$.
I've left out many important points:


*

*What is a natural number?

*What is the precise definition of the "number of elements" in a given finite set $A$? A related question is of course what a finite set is.

*How do I know that the sets $A \cup B$ and $A \times B$ are finite?

*How do I know that, if I replace $A$ and $B$ with other sets having the same numbers of elements, the number of elements in $A \cup B$ (or $A \times B$) won't change?
These are all questions tied to the foundations of mathematics, and a "proof," in order to be entirely valid, would have to be technical. However, so long as the sets $A \cup B$ and $A \times B$ are established to be finite, there can be no question that the "number of elements" of these sets will be a natural number.
You can learn more about these questions by reading about the mathematical concept of "cardinality."
A: What is the extent of the mathematical knowledge of your friend? That these statements are true are a consequence of the axiomatic construction of the natural numbers using the successor function. To say a little more, essentially, what makes $a$ and $b$ natural numbers? They are formed by adding $1$ to $1$ some number of times. Intuitively, one can see, then, that $a+b$ must be $1$ added to $1$ some number of times. This can be made rigorous using the successor function. 
Successor Function
Once you prove that $a+b$ is a natural number, proving that $i\cdot j$ for $i,j \in \mathbb{N}$ can be done inductively using the definition of $i \cdot j$, and the fact that $a+b$ is a natural number. 
