Well-Ordering Property of N

One of the most challenging theorems in Mathematics, for me, is the well-ordering property of N. You may wonder why, because it is not so hard to understand. Yes you are right, it’s easy! the principle by itself is easy to understand and also you will very soon learn how to apply it. I have this problem with proves which has been presented in different books. As you know it appears, as its nature, in very first pages of any book so called “first course in analysis”, “first course in abstract algebra” and also all books of “first course in set theory”. So any math student faces with this principal almost three times in the first year of licence. And then, as you may have seen, there is always a unique proof in each book. I have always been facing with this situation that they use something to prove a principal in which those facts by themselves more need proofs than the principle. Have you ever thought about it? why everybody insist to prove it in a new way? and why they use unclear thing to prove it? why while proving all other theorems of mathematics, all books follow the same methods?
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