>Real numbers are defined as the limits of convergent series
>of rational numbers.
>
> A = lim(n->oo){sum(i=1,n,a_i/10^i)}, with a_i =
>0,1,...,or 9.
>
>Defines a number in [0,1].
>
>To expand,
>
> For all epsilon > 0, there exists N(epsilon) such that n
>> N(epsilon) implies |sum(i=1,n,a_i/10^i) - A| < epsilon.

This is hard. I'd like to see a systematic development. E.g. I am sure if you extend Q to include all Cauchy sequences then somewhere along the way you'll have to define the operations in the extended field. But unless this is done, there is an ambiguity which is probably a source of your inquiry.

>Two questions.
>
>1) What is the meaning of |sum(i=1,n,a_i/10^i) - A| ? The
>difference between a rational number "sum(i=1,n,a_i/10^i)"
>and a number that is being defined "A" ?

So I think the right definition of a new "A" would be as a set of all Cauchy sequences that have the same limit (i.e., remain Cauchy if interwined.)

>2) What number system does epsilon come from ? The rational
>numbers or the number system we are defining ? If the
>rational numbers, then why not use 1/m for some m > 0 ?

You can do with 1/m. I do not know why the use anything else. In the definition you have to use rational numbers. But once the definition is accepted and properties of the extended operations established you can use epsilon from the extended field.