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?
 A: 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?
A: The question is foundational and thus requires a carefully definition of the concepts it is about. So, one needs to first define what integer means, and then define what the product of two such integers is. Now, you can't make an omelette without breaking eggs, so we'll have to use some prior knowledge of something. So, let's take it that we believe the natural numbers $\mathbb N$ are ok. That is, we know what they are and we know how to add and multiply them etc. 
Let $X=\mathbb N\times \mathbb N$ and define a the relation $~$ on $X$ as follows. $(x,y)~(a,b)$ precisely when $x+b=a+y$. This is defined purely in terms of natural numbers, so it's ok. It's easy to verify that this is in fact an equivalence relation. The quotient set $X/~$ is what we take as the definition of $\mathbb Z$, the integers. Thus, an integer is an equivalence class $[(a,b)]$ (which, you should convince yourself, is representing the 'usual integer' $a-b$. 
Now, we can define the product of integers: $[(x,y)]\cdot [(a,b)]=[(xa+by,xb+ay)]$. It needs to be checked that this is well-defined, i.e., that it does not depend on representatives, and it's rather straightforward to do that. This proves that the product of two integers, as we just defined those concepts, is an integer. More interestingly, one can further define all of the algebraic structure of $\mathbb Z$ as a ring, and prove all of those properties rigorously, by giving similar such definitions. 
Of course, we relied on pre-knowledge of the natural numbers, so you might ask about a proof for their properties. That can also be done using a bit of set theory. But then we'll be using properties of sets, so you might want to see a proof of their properties, in which case I'd say "welcome to axiomatic set theory".
