On page 44 of this book an approach to constructing the real numbers as equivalence classes of nested rational intervals is outlined and attributed to Bachmann. The outline in the book is very rudimentary, in fact avoiding presenting the definition of the arithmetic operations and order in favour of comparing the constructions with the other more familiar ones.
So, can the reals be constructed as equivalence classes of nested rational intervals? if so is there a reference to the construction? and how painful are the details?