Science:Math Exam Resources/Courses/MATH220/December 2009/Question 09 (b)
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Question 09 (b)
Let A be a non-empty proper subset of (0,3). Note . Define
(i) Prove that and exist using the Completeness Axiom.
(ii) Prove that is an upper bound for B.
(iii) Prove that .
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(i) The Completeness Axiom says that any nonempty set of real numbers must contain an infimum and a supremum which is exactly what we want to show.
(ii) Assume towards a contradiction that there exists an element b of the set B that is larger than the claimed supremum, i.e. . Now, by definition of B, there exists an element a inside A such that . Thus,
and clearing negatives gives
which contradicts the definition of the infimum. Hence is an upper bound of B.
(iii) Now, to show that is the least upper bound, assume towards a contradiction that there is a real number c such that c is a smaller upper bound for B. Thus . Multiplying by negative one gives
and so by definition of the infimum of A, there is an element a of A such that
Taking the negatives again gives
However this element lives inside B which contradicts the definition of c as an upper bound for B! Thus bust be the least upper bound (the supremum).