Real numbers as decimals I'm looking for a book that develops the theory of real numbers in a rigorous way in terms of their decimal expansions. The exposition should be concrete and preferably aimed at mathematically talented high-schoolers. For example, it should be pitched at readers who haven't necessarily heard of the least upper bound property prior to reading this part of the text in question.
I'd be curious to hear about books in other languages as well.
Edit. To give you a better idea of what I'm talking about, I'm including an excerpt from the  algebra textbook by Kiselev, where as an example he illustrates the meaning of $\sqrt{3} + \sqrt{2}$.

My translation of the part at the bottom:

Adding numbers $\alpha$ and $\beta$ means finding a third number $\gamma$ which is greater than the sum of any approximations from below of these numbers, yet less than the sum of any approximations of them from above. 
We accept without proof that for any two real numbers $\alpha$ and $\beta$, one and only one such number $\gamma$ exists.

Now this book doesn't actually answer my question, since it doesn't prove this statement or others like it. I am looking for something that actually carries these proofs out and makes an effort to be as accessible as possible while remaining rigorous.
 A: Page 505 of Spivak's Calculus (the very final problem in the chapter on "Construction of the real numbers") contains an exercise that begins
"This problem outlines a construction of "the high-school student's real numbers." We define a real number to be a pair $(a, \{b_n\})$ where $a$ is an integer and $\{b_n\}$ is a sequence of natural numbers from $0$ to $9$, with the proviso that the sequence is not eventually $9$; intuitively, this pair represents
$$
a + \sum_{n=1}^\infty b_n 10^{-n}.
$$
With this definition, a real number is a very concrete object, but the difficulties involved in defining addition and multiplication are formidable $\ldots$"
He then outlines a program for defining addition and multiplication and proving their properties, one that can be carried out by a bright high-school student, albeit one that has read the rest of this book, and knows things about sequences and least upper bounds, etc. He notes, wisely, that the description of mutliplicative inverses is no fun at all. 
That's hardly a "text", but it's reasonable, especially if you want to show it to very bright high schoolers. If you want to show it to others, I think you'll find that they mostly don't appreciate it, but perhaps I'm too cynical. 
A: If you can read German, here is a detailed development of the approach suggested by Gowers (see the  link given by Mark Bennet), albeit in terms of binary fractions:
http://www.math.ethz.ch/~blatter/Dualbrueche_2.pdf
But note the following: Whichever approach you take, the amount of work to be done in order to verify all the details is about the same.
