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Cauchy sequences of rationals can be used to model the reals.

Has anyone actually tried though to develop a theory of real analysis explicitly in this model?

Some definitions seem like they would be straightforward to convert into this model, but I am not sure about others.

I guess we probably wouldn't gain anything, but it would be interesting to see.

Does someone have a reference maybe?

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I wonder why you would want to do this? Since you have to take equivalence classes of the Cauchy sequences, which essentially gives you the reals, there's nothing to gain here. I mean, the reals are equivalence classes of Cauchy sequences, just with a more compact notation. –  pki Nov 6 '11 at 15:11
    
Yes, I know that. The only thing that might be gained is insight. It just seemed interesting to me. No other reason. –  Tim Seguine Nov 6 '11 at 16:49
    
"Some definitions seem like they would be straightforward to convert into this model" - do you have any examples? –  Srivatsan Nov 6 '11 at 16:58
    
@Srivatsan Like for example, I could define the exponential function of a Cauchy sequence as $\exp{\{x_n\}}_{n\in\mathbb{N}}:=\{\sum_{k=0}^{n}\frac{x_n^k}{k!}\}_{n\in\mathbb‌​{N}}$ –  Tim Seguine Nov 6 '11 at 17:40
    
@pki The reason I originally came up with the idea was for discretizing algorithms. I thought maybe by manipulating the series directly, one might find explicit representations for the series that converges to the solution. Seems probably impractical now, but the idea is still intriguing to me. –  Tim Seguine Nov 6 '11 at 18:04

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