Exercise 2.6.6. (a) Assume the Nested Interval Property (Theorem 1.4.1) is
true and use a technique similar to the one employed in the proof of the Bolzano– Weierstrass Theorem to give a proof for the Axiom of Completeness. (The reverse implication was given in Chapter 1. This shows that AoC is equivalent to NIP.)
(b) Use the Monotone Convergence Theorem to give a proof of the Nested
Interval Property. (This establishes the equivalence of AoC, NIP, and MCT.)
(c) This time, start with the Bolzano–Weierstrass Theorem and use it to
construct a proof of the Nested Interval Property. (Thus, BW is equivalent to NIP, and hence to AoC and MCT as well.)
(d) Finally, use the Cauchy Criterion to prove the Bolzano–Weierstrass The-
orem. This is the ﬁnal link in the equivalence of the ﬁve characterizations of completeness discussed at the end of Section 2.6.
This is from Stephen Abbott's "Understanding Analysis". I've attached chapter 2 from the book.
Jun 1st 2010, 12:17 PM
What have YOU done on this? What are your ideas?
Jun 2nd 2010, 03:19 PM
I worked on this problem for two days and went to my teacher for help, but he said my work was garbage and told me to start over after throwing my paper away. I am not looking for anyone to hand me the answer. I just want someone to help me get started or to point me in the right direction, especially on (c). Any help would be appreciated.
Jun 3rd 2010, 10:54 AM
I talked with a different teacher and I think I've got it now.