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).

(* We can probably reduce the factor 4. *)
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).