I wrote:

Does anyone know of a good write-up of the construction of the reals and their arithmetic via decimal expansions?

The only example I can think of is F. Faltin, N. Metropolis, B. Ross and G.-C. Rota, The real numbers as a wreath product, Advances in Math., 16(1975), 278--304.

I should mention that one of my own writing projects over the next few years is to write an article that does this "the right way", mostly following in the lines of Faltin et al. (but with some of the bumps smoothed away), give it a title less intimidating than "The real numbers as a wreath product", and submit it to the Monthly rather than Advances in Math. But I'd be pleased to find out that someone else has already written such an article so that I don't have to! (For instance, in writing such an article I'd probably have to check out what Weierstrass did when he constructed the reals via radix expansions, and I suspect it'll be tough reading.) For those of you who haven't seen the Faltin et al. article, let me mention just one of their tricks: Allow the digits in a base r expansion to be arbitrary non-negative integers, not just elements of {0,1,...,r-1}, and then introduce real numbers as equivalence classes of such expansions. Jim Propp