(** Continuity of norm *)

(* Specifications: *)
Lemma filterlim_norm {K : AbsRing} {V : NormedModule K} :
forall (x : V), filterlim norm (locally x) (locally (norm x)).
Lemma filterlim_norm_zero {U} {K : AbsRing} {V : NormedModule K}
{F : (U -> Prop) -> Prop} {FF : Filter F} (f : U -> V) :
filterlim (fun x => norm (f x)) F (locally 0)
-> filterlim f F (locally (zero (G := V))).
Lemma filterlim_bounded {K : AbsRing} {V : NormedModule K} (a : nat -> V) :
(exists x, filterlim a eventually (locally x))
-> {M : R | forall n, norm (a n) <= M}.