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So I've gotten myself confused over a seeming tautology one uses when constructing the real numbers as equivalence classes of Cauchy sequences of rational numbers. Having constructed the real numbers in this way, once one makes sense of the term "positive real number" it is easy to extend the absolute value and in particular get a notion of convergence.

If one has a Cauchy sequence of rational numbers which represents the class of some real number, surely this sequence converges to this number, but how does one actually see this without begging the question?

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    An important point. After constructing the reals after all the trouble, you should prove that a Cauchy sequence of *reals* does in fact converge to a real number. If this is not satisfied, you will have to repeat the process of inventing more numbers for this purpose.2011-09-10
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    @Srivatsan: right, and I'm asking about an even more basic fact, which is that any Cauchy sequence of rational numbers converges to a real number. This, together with the density of the rationals in the reals, implies the completeness of the real numbers.2011-09-10
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    @Srivatsan: Slightly off at a tangent, but I think that it's an interesting fact that in constructive logic, the reals constructed in this way are *not* Cauchy complete.2011-09-10
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    @George I didn't know that. Thanks :)2011-09-10
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    @Srivatsan: I should back up that statement with a link. Although it's not where I originally learned this from, a google search came up with the following. *On the Cauchy Completeness of the Constructive Cauchy Reals* http://math.fau.edu/lubarsky/Cauchyreals.pdf2011-09-10

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