- #1
saim_
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I need help with the last part of Problem 20, Chapter 1 of Rudin. Here's the problem:
"With reference to the Appendix, suppose that property (III) were omitted from the definition of a cut. Keep the same definitions of order and addition. Show that the resulting ordered set has the least-upper-bound property, that addition satisfies axioms (A1) to (A4) (with a slightly different zero-element!) but that (A5) fails."
I'm just having trouble proving why A5 would fail without property (III).
Note: The appendix, referred to here, contains construction of reals using Dedekind cuts; property (III) is the requirement that a cut have no largest number; properties A1 to A4 are field axioms of closure, associativity, commutativity and existence of identify for addition. A5 is the property of existence of an additive inverse.
"With reference to the Appendix, suppose that property (III) were omitted from the definition of a cut. Keep the same definitions of order and addition. Show that the resulting ordered set has the least-upper-bound property, that addition satisfies axioms (A1) to (A4) (with a slightly different zero-element!) but that (A5) fails."
I'm just having trouble proving why A5 would fail without property (III).
Note: The appendix, referred to here, contains construction of reals using Dedekind cuts; property (III) is the requirement that a cut have no largest number; properties A1 to A4 are field axioms of closure, associativity, commutativity and existence of identify for addition. A5 is the property of existence of an additive inverse.