Definition CReal_from_cauchy_seq (xn : nat -> CReal) (xcau : Un_cauchy_mod xn) (n : Z) : Q :=
  let p := CReal_from_cauchy_cm n in
  let (q, _) := xcau (4 * 2^p)%positive in
  seq (xn q) (Z.neg p - 2)%Z.

Lemma CReal_from_cauchy_cauchy : forall (xn : nat -> CReal) (xcau : Un_cauchy_mod xn), QCauchySeq (CReal_from_cauchy_seq xn xcau).