I have tried to explain that finite calculational accuracy is the point: it is not possible to calculate real number exactly in finite time and no-one has been claiming anything like that. The idea of giving up all the mathematics since Newton is just just because we cannot calculate with infinite precision is complete idiotism.

And I am still unable to see what is wrong with Cauchy sequences: here I tried to concretise them in terms of decimal representation in order to give the idea what they are about but it seems that it did not help.

The generalisation of real numbers rather than refusing to admit their existence, is the correct direction to proceed and I have been working with this problem with strong physical motivations. Fusion of reals and p-adics to adelic structures also at space-time level, hierarchy of infinite primes defining infinite hierarchy of second quantisation for an arithmetic quantum field theory, even construction of arithmetics of Hilbert spaces, replacement of real point with infinitely structured point realizing number theoretic Brahman = Atman/algebraic holography. These are interesting lines to proceed rather than a return to cave.