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 CRealLt_QR_from_single_dist : forall (q : Q) (r : CReal) (n :Z),
(2^n < seq r n - q)%Q
-> inject_Q q < r .