Reading: Syllabus. Text: Chapter 1 through the Extended Real Number System and skim the Appendix to Chapter 1.
Problems: Chapter 1: 1,2,4,5,7 (Assume Exercise 6, in which bx is constructed).
Problems A and B below aim at showing that Q
does not have the least upper bound property and deal with fine-tuning
the argument of Theorem 1.21 in the simple case of the nth root
of 12 for n = 2, so that it works for the set of
positive rationals whose square is less than 12 in lieu of the
set of positive reals whose square is less than 12. Do not
assume that the
Problem A: Show that if x > 0 is real and x2 < 12, then there is a rational y > x such that y2 < 12. Hint: Show that if h > 0 is a sufficiently small, then (x + h)2 < 12. Then use Theorem 1.20(b).
Problem B: Show that the set of positive rational numbers x such that x2 < 12 does not have a least upper bound in Q. You may use Exercise 2 from the text.
Problem C: Prove that every ordered set with the greatest lower bound property also has the least upper bound property.
Last modified: (2014-09-13 09:17:37 CDT)