Section fct_CompleteSpace.
Context {T : Type} {U : CompleteSpace}.

Lemma filterlim_locally_cauchy :
forall {F} {FF : ProperFilter F} (f : T -> U),
(forall eps : posreal, exists P, F P /\ forall u v : T, P u -> P v -> ball (f u) eps (f v)) <->
exists y, filterlim f F (locally y).