I'm a very newbie, even if I studied some math at college... Now I'm trying to study "Contemporary Abstract Algebra", by J.A. Gallian but I'm already facing some difficulties trying to understand the proof of the very first theorem (0.1) about division algorithm.

The theorem says: 

> 
> ￼Let a and b be integers with $b > 0$. Then there exist unique integers
>  q and r with the property that $a=bq + r$, where $0\le r\le b$
> 
> 
> 

The proof starts in this way:

> 
> We begin with the existence portion of the theorem.
> 
> 
> 

Ok, what does that mean?
Moreover:

> 
> Consider the set S = {$a - bk | k$ is an integer and $a - bk\ge 0$}.
> 
> 
> 

Ok, what are we talking about? What is k? Why doesn't he use q?
What kind of set is this? Is a set with just this "difference" between a and bk? Why?

Shall I ever be able to understand this?

Thanks a lot