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By Hutchinson J.

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In particular, we can solve simultaneous linear equations. We will also assume standard 2 definitions including x2 = x × x, x3 = x × x × x, x−2 = (x−1 ) , etc. 2 Important Sets of Real Numbers We define 2 = 1 + 1, 3 = 2 + 1 , . . , 9 = 8 + 1 , 10 = 9 + 1 , . . , 19 = 18 + 1 , . . , 100 = 99 + 1 , . . The set N of natural numbers is defined by N = {1, 2, 3, . }. The set Z of integers is defined by Z = {m : −m ∈ N, or m = 0, or m ∈ N}. The set Q of rational numbers is defined by Q = {m/n : m ∈ Z, n ∈ N}.

A real number is irrational if it is not rational. 1, the real numbers have a natural ordering. Instead of writing down axioms directly for this ordering, it is more convenient to write out some axioms for the set P of positive real numbers. We then define < in terms of P . Order Axioms There is a subset6 P of the set of real numbers, called the set of positive numbers, such that: A10 For any real number a, exactly one of the following holds: a = 0 or a ∈ P or −a∈P A11 If a ∈ P and b ∈ P then a + b ∈ P and ab ∈ P A number a is called negative when −a is positive.

B 0 ( T -b ) S Since S is bounded below, it follows that T is bounded above. b. b. for S. 12 It follows that S has infinitely many upper bounds. 28 Equivalence of A12 and A12 1) Suppose A12 is true. We will deduce A12 . For this, suppose that S is a nonempty set of real numbers which is bounded above. 13 Note that B = ∅; and if x ∈ S then x − 1 is not an upper bound for S so A = ∅. The first hypothesis in A12 is easy to check: suppose a ∈ A and b ∈ B. If a ≥ b then a would also be an upper bound for S, which contradicts the definition of A, hence a < b.

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Introduction to Mathematical Analysis by Hutchinson J.


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