I don't want to be disagreeable, but it seems clear to me that this construction gives Cauchy reals, not Dedekind reals and only the latter can be said to go back to Eudoxus. Indeed, given a nearly function f, with bound B on the near linearity, it is easy to see that |(fn/n) - (fm/m)| =< B(1/n + 1/m) so that the sequence fn/n is Cauchy. Michael