Definition QCauchySeqLin (un : positive -> Q)
  : Prop
  := forall (k : positive) (p q : positive),
      Pos.le k p
      -> Pos.le k q
      -> Qlt (Qabs (un p - un q)) (1 # k).

(* Specifications: *)
Lemma Rcauchy_limit : forall (xn : nat -> CReal) (xcau : Un_cauchy_mod xn), QCauchySeqLin (fun n : positive => let (p, _) := xcau (4 * n)%positive in seq (xn p) (4 * Z.neg n)%Z).