They are. To get enumerations of rationals above and below out of an effective Cauchy sequence, once the Cauchy sequence outputs a rational q such that everything afterwards can only differ by at most ε, you start enumerating rationals below q−ε as below the real and rationals above q+ε as above the real. If the Cauchy sequence converges to r, and you have a rational q<r, then once the Cauchy sequence gets to the point where everything after is gauranteed to differ by at most r−q2, you can enumerate q as less than r.
They are. To get enumerations of rationals above and below out of an effective Cauchy sequence, once the Cauchy sequence outputs a rational q such that everything afterwards can only differ by at most ε, you start enumerating rationals below q−ε as below the real and rationals above q+ε as above the real. If the Cauchy sequence converges to r, and you have a rational q<r, then once the Cauchy sequence gets to the point where everything after is gauranteed to differ by at most r−q2, you can enumerate q as less than r.