The problem here is a confusion of "Cauchy sequence" used in real analysis, where the "Cauchy property" implies limit. And the "Cauchy sequence" in foundational construction of the reals. Where you don't have any limits available,
because you are just about to construct the reals. I admit, it needs a little bit elasticity of the brain, a big problem for bird brains like John Gabriel. It wouldn't be a problem if his brain had a switch, which reads as follows: - Switch State 1: We doing foundational math - Switch State 2: We are doing real analysis
Of course the are not seemless State 1 and State 2. We wanted to explain this to you already in the case of the rational numbers. Foundational they are pairs <p,q> representing fractions p/q.
But if you work with Q, these pairs disappear. Its like with "S=Lim S", the difference disappears, its a lot of magic for a bird brain, and very different from cheese rolling, pitty you didn't win:
Am Mittwoch, 4. Oktober 2017 21:18:25 UTC+2 schrieb burs...@gmail.com: > But you don't need the value zero, you indentify > those sequences that have property Z with zero. > > Same with other limits, you don't have them as > value, you only have the sequences itself. > > Am Mittwoch, 4. Oktober 2017 21:16:21 UTC+2 schrieb burs...@gmail.com: > > You see bird brain John Gabriel doesn't understand how > > the reals are constructed in the case of Cauchy Q-series. > > > > Some weeks ago I posted a PDF from cornell university, > > which showed the construction of reals from Q-series, > > > > MATH 304: CONSTRUCTING THE REAL NUMBERS > > Peter Kahn Spring 2007 > > http://www.math.cornell.edu/~kahn/reals07.pdf > > > > here it is again. What Markus Klyver wrote is correct, > > check it by yourself, the construction involves the > > > > concept of so called Z series. > > 4.3 The Field of real numbers > > "Z is defined to consist of all sequences > > in C that converge to zero." > > > > Am Mittwoch, 4. Oktober 2017 21:08:51 UTC+2 schrieb John Gabriel: > > > Wrong. All limit definitions require prior knowledge of the limit. This is especially true in the case of the derivative. > > > > > > > We can perfectly define limits without knowing how to prove limits or evaluating limits. > > > > > > Wrong. > > > > > > > > > > > And no, we don't define real numbers as limits of Cauchy sequences. > > > > > > Yes, you do! All the sequences of an equivalent Cauchy sequence have one thing in common - the limit.