I understand what you're saying and I think it's a good point. The problem is that you're developing an algorithm (a non-terminating one) that finds real numbers rather than providing a definition of them. It turns out that providing a definition of real numbers is not a simple as it may at first seem. This presentation is somewhat similar constructive analysis, in which a real number is defined as regularly converging sequence of rational numbers; importantly, constructive analysis does not define real numbers as *infinite sums* of these sequences, because as I've said, that would be a circular definition.

If you want to learn more about rigorous foundations for real numbers and related topics, I think that the book Calculus by Michael Spivak is a very approachable and well respected introduction to the topic.