# Proof of the division algorithm

#### matqkks

##### New member
In many books on number theory they define the well ordering principle (WOP) as:
Every non- empty subset of positive integers has a least element.
Then they use this in the proof of the division algorithm by constructing non-negative integers and applying WOP to this construction. Is it possible to apply the WOP to a subset of non-negative integers? Am I being too pedantic?

#### Dr.Peterson

##### Elite Member
Since the principle as you state it requires positive integers, it can't be applied directly.

But given a set $$\displaystyle A$$ of non-negative integers, you could form the set $$\displaystyle B = \{a+1 : a\in A\}$$, and apply the principle to that.

Thanks.