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Exercises 2.4.4 and 2.5.4 establish the equivalence of the Axiom of Completeness and the Monotone Convergence...

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Advance Math

Exercises 2.4.4 and 2.5.4 establish the equivalence of the Axiomof Completeness and the Monotone Convergence Theorem. They alsoshow the Nested Interval Property is equivalent to these other twoin the presence of the Archimedean Property.

(a) Assume the Bolzano-Weierstrass Theorem is true and use it toconstruct a proof of the Monotone Convergence Theorem withoutmaking any appeal to the Archimedean Property. This shows that BW,AoC, and MCT are all equivalent.

(b) Use the Cauchy Criterion to prove the Bolzano-WeierstrassTheorem, and find the point in the argument where the ArchimedeanProperty is implicitly required. This establishes the final link inthe equivalence of the five characterizations of completenessdiscussed at the end of Section 2.6.

(c) How do we know it is impossible to prove the Axiom ofCompleteness starting from the Archimedean Property?

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