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Math Help - Cauchy Criterion

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    Cauchy Criterion

    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.
    Attached Files Attached Files
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    What have YOU done on this? What are your ideas?
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  3. #3
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    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.
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  4. #4
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    I talked with a different teacher and I think I've got it now.
    Last edited by hejupr; June 3rd 2010 at 06:35 PM. Reason: found correct answer, don't need help
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