I'm working through Pinter's "A Book of Abstract Algebra" and something is bothering me about his use of the well-ordering property to produce a non-negative remainder.
He writes:
"Let $W$ be the subset of $\mathbb{Z}$ consists of all nonnegative integers which are expressible in the form $m - xn$, where $x$ is any integer. By (a previous argument), $W$ is not empty; hence by the well-ordering property, $W$ contains a least integer $r$. Because $r \in W$, $r$ is nonnegative and is expressible in the form $m-nq$ for some integer $q$.
Thus, we already have $m = nq + r$ and $0 \leq r$. It remains only to verify that $r < n$."
and then he goes on to show that the minimality of $r$ would be violated if $ r \geq n$.
My problem is in using the well-ordering property to obtain $r$ in the case that the remainder is actually 0. For example, if $ m = 12$ and $ n = 4$, then the set $W = \{0, 4, 8, ...\}$, and the well-ordering property gives us that $r = 4$. Now it is the case that $ r \geq 0$, but $r$ does not satisfy $ r < n$. Of course, there $is$ an $r$ that does (0), but the well-ordering property can't give it to us.
So I feel that this proof is incomplete, in that this case is not handled properly.
Am I missing something subtle or is this really just an oversight?
