Why Read This Book? - Index of

58 Chapter 2 Properties of Real Numbers
(A19) Well-ordering principle: Any non-empty subset of whole numbers has a
smallest element. That is, if A is non-empty, then there is some number
a ∈ A with the property that a ≤ x for all x ∈ A.
Theorem 2.4.7 (Division Algorithm). Let a and b be integers, where a>0.
Then there exist unique integers q and r such that b = aq + r and 0 ≤ r0 and b ≥ 0, we can apply the WOP
to S, then show that this smallest element is the value of r we’re after. Here is the
proof of the existence part of Theorem 2.4.7, in as much detail as we promised.
Proof. (Existence) Let a and b be integers, where a>0. First, we consider the
case b ≥ 0. Define the set S as in Eq. (2.14). By definition, S ⊆ W, and since b ≥ 0,
we may let q = 0 to see that b ∈ S, so that S is non-empty. By the WOP, S has a
smallest element, which we may denote r, and note that r is of the form b − aq
for some integer q. Thus we have that b = aq + r, where r ≥ 0, and we must show
that r