Definition series_cv {R : ConstructiveReals}
(un : nat -> CRcarrier R) (s : CRcarrier R) : Set
:= CR_cv R (CRsum un) s.
Definition series_cv_lim_lt {R : ConstructiveReals}
(un : nat -> CRcarrier R) (x : CRcarrier R) : Set
:= { l : CRcarrier R & prod (series_cv un l) (l < x) }.
Definition series_cv_le_lim {R : ConstructiveReals}
(x : CRcarrier R) (un : nat -> CRcarrier R) : Set
:= { l : CRcarrier R & prod (series_cv un l) (x <= l) }.

(* Specifications: *)
Lemma series_cv_maj : forall {R : ConstructiveReals}
(un vn : nat -> CRcarrier R) (s : CRcarrier R),
(forall n:nat, CRabs R (un n) <= vn n)
-> series_cv vn s
-> { l : CRcarrier R & prod (series_cv un l) (l <= s) }.
Lemma series_cv_abs_lt
: forall {R : ConstructiveReals} (un vn : nat -> CRcarrier R) (l : CRcarrier R),
(forall n:nat, CRabs R (un n) <= vn n)
-> series_cv_lim_lt vn l
-> series_cv_lim_lt un l.