Definition talk:Real Number
Circular problems with the definition using Cauchy Sequences
Many articles about sequences assume we have defined real numbers. Examples: Definition:Convergent Sequence/Note on Domain of N, Definition:Metric Space.
If we don't want to invalidate the construction of $\R$ by Cauchy-sequences, we have to make sure that all definitions and theorems used in its construction do not use $\R$ (better: don't even mention it). This means: when defining an equivalence relation on Cauchy Sequences, this has to be done separately for rational numbers. The theorem for general metric spaces can not be used in the construction of $\R$. --barto (talk) 07:49, 28 January 2017 (EST)