A draft written in Italian proves the existence and uniqueness of an element that separates two non-empty sets representing the field of real numbers.
This allows us to prove that the real numbers are a complete space (i.e., that every Cauchy sequence is convergent).
Note: in the case of irrational numbers instead of rational numbers, it does not work because they are not complete, i.e., it is not possible to define an element that is an extreme.
In layman’s terms, one could say that if rational numbers are removed from the real numbers to obtain irrational numbers, then ‘gaps remain’ and completeness no longer exists.
upgrade: a reconsideration of the demonstration
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