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.