# Number TheoryProof of the Division Algorithm

#### matqkks

##### 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?

#### HallsofIvy

##### Well-known member
MHB Math Helper
It's rather obvious isn't it? "Applying the WOP to a subset of non-negative integers" would simply mean that, given X, a subset of the non-negative integers, any subset of X has a least member. And that is true because any subset of X is also a subset of the non-negative integers.

If that is not what you mean then please explain what you mean by "apply the WOP to a subset of non-negative integers".

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#### matqkks

##### Member
Yes of course. I just had a senior moment.
Thanks.

#### HallsofIvy

##### Well-known member
MHB Math Helper
There are those of use who live in "senior moments"! We are called "seniors".

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