Section Definitions.
(** * Definitions *)
Context {K : AbsRing} {V : NormedModule K}.
Definition is_series (a : nat -> V) (l : V) :=
filterlim (sum_n a) (eventually) (locally l).
Definition ex_series (a : nat -> V) :=
exists l : V, is_series a l.
Definition Cauchy_series (a : nat -> V) :=
forall eps : posreal,
exists N : nat, forall n m : nat,
(N <= n)%nat -> (N <= m)%nat ->
norm (sum_n_m a n m) < eps.
End Definitions.
Definition Series (a : nat -> R) : R :=
real (Lim_seq (sum_n a)).
(** Cauchy *)

Lemma Cauchy_ex_series {K : AbsRing} {V : CompleteNormedModule K}
(a : nat -> V) :
ex_series a -> Cauchy_series a.