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PART I. THE REAL NUMBERS - UH

PART I. THE REAL NUMBERSThis material assumes that you are already familiar with the real number system and the represen-tation of the real NUMBERS as points on the real THE NATURAL NUMBERS AND INDUCTIONLetNdenote the set of natural NUMBERS (positive integers).Axiom:IfSis a nonempty subset ofN, thenShas a least element. That is, there is anelementm Ssuch thatm nfor alln :A set which has the property that each non-empty subset has a least element is said to bewell-ordered. Thus, the axiom tells us that the natural NUMBERS are a subset the following properties:1. 1 S, Simpliesk+1 S,thenS= :SupposeS6=N. LetT=N S. ThenT6= . Letmbe the least element inT. Thenm 1/ T. Therefore,m 1 Swhich implies that (m 1) + 1 =m S, a :LetSbe a subset ofNsuch Ifk m S, thenk+1 ,S={n N:n m}.ExampleProve that 1 + 2 + 22+23+ +2n 1=2n 1 for alln the set of integers for which the statement is 20=1=21 1, 1 that the positive integerk +21+ +2k 1+2k=(20+21+ +2k 1)+2k=2k 1+2k=2 2k 1=2k+1 ,k+1 have shown that 1 Sand thatk Simpliesk+1 S.

Other examples of irrational numbers are √ m where m is any rational number which is not a perfect square, 3 √ m where m is any rational number which is not a perfect cube, etc. Also, the numbers π and e are irrational. Definition 2. Let S be a subset of R. A number u ∈ R is an upper bound of S if s ≤ u for all s ∈ S .

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