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# Real and complex analysis

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... You can explain it to average calculus students, and even lead them to conjecture it on their own; the only thing that's hard is convincing them that it's nontrivial! The Cut Property hasn't been entirely forgotten ( [1], [16, p. 53], and [20]) and it's well-known among people who study the axiomatization of Euclidean geometry [10] or the theory of partially ordered sets and lattices [28]. But it deserves to be better known among the mathematical community at large. ...
... The Ratio Test (16) implies completeness by way of the Archimedean Property (2) and the Cut Property (3): Note that the Ratio Test implies that 1 2 + 1 4 + 1 8 + . . . converges, implying that R is Archimedean (the sequence of partial sums 1 2 , 3 4 , 7 8 , . . . isn't even a Cauchy sequence if there exists an ǫ > 0 that is less than 1/n for all n). ...
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