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Well ordering principle and Archimedean property

Well ordering principle: Every non empty set S of non-negative integers contains a least element; that there is some integer a in S such that a≤b for all belonging to S.

Theorem(Archimedean property): If a and b are any positive integers then there exists a positive integer n such that na≥b.

Proof: Assume that the statement of the theorem is not true, so that for some a and b, na<b for every positive integer n. Then the set S={b-na|na positive integer}consists entirely positive integers.

By the well ordering principle S will possess a least element.

Say b-ma we notice that b-(m+1)a also lies in S. Since S contains all the integers of this form. Furthermore, we have b-(m+1)a=(b-ma)-a<b-ma, contry to the choice of b-ma as the smallest integer in S. This contradiction arose out of our original assumption that the Archimedean property died not held.

Hence the property is true.(Proved)

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