Here is a minimalistic theory of real numbers, all the proofs
are left to the reader as exercises.

We assume the finite decimals as known. A positive real number is 
an infinite decimal extending indefinitely to the right.
More formally, it is a pair consisting of a finite sequence
of digits and an infinite sequence of digits, the first representing
the integral part and the second -- the fractional part. 
A real number consists of a positive real number and a sign.
By an infinte sequence we understand a function on N 
(= the natural numbers). We assume the usual rounding of 9999... tails.


1) Rounding: 5,6,7,8,9 --> 10, 0,1,2,3,4 -->0, 
[a]_k is the rounding of a to k decimal places.

2) 2 real numbers a and b are the same 
iff for any k [a]_k=[b]_k or [a]_{k+1}=[b]_{k+1}

3) |[a]_{k+m} - [a]_k|<=10^{-k} 

4) For any fixed n one of the sequences 
[[a]_k+[b]_k]_n or [[a]_k+[b]_k]_{n+1} stabilizes for k>K(n),
the same for -,* and / (if b is not 0) instead of +.
We take it as the definition of the arithmetical operations. 

5) Definition of the limit: a_k --> a iff for any n
[a_k]_n stabilizes to [a]_n or  [a_k]_{n+1} stabilizes to [a]_{n+1}.

6) Theorem: If {a_k} is Cauchy then for any n
[a_k]_n stabilizes or [a_k]_{n+1} stabilizes,
the limit of {a_k} is a with the corresponding stable digits.