It is consistent with constructive set theory (without Countable Choice,
clearly) that the Cauchy reals (equivalence classes of Cauchy sequences of
rationals) are not Cauchy complete. Related results are also shown, such as
that a Cauchy sequence of rationals may not have a modulus of convergence, and
that a Cauchy sequence of Cauchy sequences may not converge to a Cauchy
sequence, among others