In article <34E95F44.5FE6@pitt.edu>, Michael David Cochran <firstname.lastname@example.org> wrote: >Does anyone have a fast proof of the Well-Ordering Property? The >Well-Ordering Property states that every nonempty set of positive >integers contains a smallest member.
Since the set S is non-empty, it contains at least one element. Choose one. For convenience, call it n. Now if n is not itself the smallest element in S, then the smallest element would be smaller than n. Hence without loss of generality we may replace S by the subset of S consisting of integers in S less than or equal to n. But there are only finitely many positive integers less than or equal to n. (In fact, there are only n of them!) Therefore, without loss of generality we may suppose that S is finite. But clearly every finite set of integers (or real numbers for that matter) has a smallest element.
If you can read pdf (Adobe Acrobat) files, you might be interested in a rather extended discussion of some related matters in my lengthy article on the Division Algorithm: www.math.Hawaii.Edu/~lee/courses/Division.pdf. (Versions of the article in dvi format and postscript format are also available.)
-- Trying to understand learning by studying schooling is rather like trying to understand sexuality by studying bordellos. -- Mary Catherine Bateson, Peripheral Visions lady@Hawaii.Edu