>For example the brainwashed moron Klyver will harp on the irrelevant fact that a Cauchy sequence of rationals may not converge to a rational number, BUT this does not mean the sequence does not converge because ALL Cauchy sequences WITHOUT ANY EXCEPTIONS converge.
Incorrect, the fact that it doens't converge to a rational number means it doesn't converge in the rationals. Therefore by definition it is DIVERGENT! You are only garantueed that convergence exists when you are dealing with a complete space, rational numbers are not complete.
You cannot even state definitions correctly.
>Notice that irrational number is defined by Cauchy HIMSELF in terms of LIMITS, NOT EQUIVALENCE CLASSES which are a NON-REMARKABLE CONSEQUENCE of limits.
Cauchy actaully never did the definition, he just said that any real number has a rational number cauchy sequence converging to it. Which others later used to make the cauchy sequence definition and to make it work they, unlike you, understood the importans of equivalence classes.