Proof that if $x, y \in \mathbb{Z}$ then $xy \in \mathbb{Z}$ How do you prove that the product of two integers is an integer?

Let's say you assume that $\mathbb{Z}$ is a group under addition, then it is just induction : If $x \in \mathbb{Z}$, then it suffices to prove that $nx \in \mathbb{Z}$ for all $n \in \mathbb{N}$, since if $nx \in \mathbb{Z}$, then $-nx \in \mathbb{Z}$.
To that end, say $n=2$, then
$$
nx = x + x  \in \mathbb{Z}
$$
so assume that $n \geq 3$, and $(n-1)x\in \mathbb{Z}$, then
$$
nx = (n-1)x + x \in \mathbb{Z}
$$
So the real question is, by your definition of a number, can you show that $(\mathbb{Z}, +)$ is a group?
