That no limits are available has Markus Klyver already explained a 3 trillion times, Q-sequences
need not have a Q-limit. So bird brain John Gabriel, maybe open a book once a while,
you know these beige things. You say its brain wash? Ok so then, stay illiterate.
Am Mittwoch, 4. Oktober 2017 21:27:46 UTC+2 schrieb burs...@gmail.com: > 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: > > Cheese Rolling 2017 at Cooper's Hill > https://www.youtube.com/watch?v=pK1j06Gjp94 > > 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.