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I know that similar questions have been asked before, but I can't seem to find something that really justifies the Cauchy construction of the reals. One question that seemed to have been asked is how can Cauchy sequences exist without the real numbers. That's fine. You can use rationals. But more fundamentally, metrics are defined as real values functions. In fact, if you defined metrics as being rational valued functions, you run into the problem that there concept of metric falls apart for the reals.

Dedekind cuts avoid all of this and honestly seem like the "correct" way to define the reals. Unless there is a way to define a metric without them.

Daniel Goldman
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  • See this answer by kahen to an earlier question asking about the construction; it’s extremely detailed, and you’ll see that you don’t need the notion of a metric. – Brian M. Scott Oct 22 '14 at 01:51
  • I see. This certainly works. It's definitely not the way that's being addressed in the course in Real Analysis that I am taking. I feel that Dedekind cuts are... much less work. – Daniel Goldman Oct 22 '14 at 01:59
  • It depends: Dedekind cuts are easier if you’re primarily interested in the order properties of $\Bbb R$, but Cauchy sequences are probably a little easier if you’re more interested in the arithmetic properties. I’d say, though, that it’s good to have at least a general understanding of both. – Brian M. Scott Oct 22 '14 at 02:04
  • Fair. I'm going to have to parse through that guy's explanation for quite a while. I somehow doubt that's how it's covered in a 500 level graduate course. – Daniel Goldman Oct 22 '14 at 02:19

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Don't bother defining 'metric'. Define 'absolute value' as a function from a field to itself satisfying such-and-such properties. Then define Cauchy sequences using the rationals and their absolute value.

(After you've got the reals, then go ahead and define metrics as real-valued functions, and define completions of arbitrary metric spaces — for which, of course, Dedekind cuts are not so useful.)