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_abs_Qabs_diff: forall (x y : CReal) (q : Q) (n : Z),
CReal_abs (x - y) <= inject_Q q
-> (Qabs (seq x n - seq y n) <= q + 2*2^n)%Q.