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. 

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. 
