Reading: Syllabus. Text (Junghenn): Chapter 1 through Section 1.4.
Problems: Section 1.2: 1(e) (assume also
that c is not 0, something just overlooked by the author of the text),
4(a), 6.
Section 1.3: 13(a,d), 16(a).
Section 1.4: 1, 4, 6(a), 10, 19.
Problems A and B below aim at showing that Q does not have the least upper bound property. Do not assume that √ 2 (or any other irrational number, for that matter) exists in the reals R. Otherwise, Problems A and B will be too simple. And, really, Problem B is just about Q: in principle, one does not need to know anything about R to solve it. In Problem A, one can start with a rational x and look for a rational h, and then Problem A also becomes a problem dealing only with the rationals. Thus, what we are doing here is this: Q is an ordered field, so that it makes sense to ask whether it possesses the least upper bound property and answer the question within the means of rational numbers.
Problem A: Show that if x > 0 is real and x2 < 2, then there is a rational y > x such that y2 < 2. Hint: Show that you can find h > 0 so small that (x + h)2 < 2. Then use Theorem 1.4.8.
Problem B: Show that the set of positive rational numbers x such that x2 < 2 does not have a least upper bound in Q.
Last modified: (2015-09-18 14:53:57 CDT)