Question

Can any genius explain me about well-ordering principle - Proofs using well-ordering principle. with some examples.

Answer 1

Statement:
Every nonempty subset of the positive integers has a least element.
Here are several examples of properties of the integers which can be proved using the well-ordering principle. Note that it is usually used in a proof by contradiction; that is, construct a set S suppose S is nonempty, obtain a contradiction from the well-ordering principle, and conclude that must be empty.
2nd part:
Theorem:There are no positive integers strictly between 0 and 1.
Proof:
Let S be the set of integers x such that 0<X<1. Suppose S is nonempty; let n be its smallest element. Multiplying both sides of n<1 by n gives n^2<n. The square of a positive integer is a positive integer, so n^2 is an integer such that 0<n^2<n<1.This is a contradiction of the minimality of n. Hence S is empty.
Another theorem:Every positive integer >1 has a prime divisor.
Proof:
Let S be the set of positive integers>1 with no prime divisor. Suppose S is nonempty. Let n be its smallest element. Note that n is cannot be prime, since n divides itself and if n were prime, it would be its own prime divisor. So n is composite: it must have a divisor d with 1<d<n.But then d must have a prime divisor (by the minimality of n). Call it p. Then p|d, but d|n ,so p|n. This is a contradiction. Therefore S is empty.