This post concerns the existence part of the completeness theorem for real numbers: "Every Cauchy sequence of real numbers has a real number as limit". Could anyone please direct me to a proof of this theorem that
- is concerned with real numbers defined as equivalence classes of Cauchy sequences, and
- proceeds directly from the definitions of "real number", "Cauchy sequence" and "limit" rather than appealing to some other "substantial" theorem (e.g. the least upper bound theorem).
I already have my hands on such a proof, namely the one in David A. Sprecher's "Elements of Real Analysis", but I don't understand it: it is badly written and, I think, full of typos. I have spend hours googling for another proof, but I can't find anything that satisfies the criteria mentioned above.