Section Locally.
Context {T : UniformSpace}.
Definition locally (x : T) (P : T -> Prop) :=
exists eps : posreal, forall y, ball x eps y -> P y.
Global Instance locally_filter :
forall x : T, ProperFilter (locally x).
Definition is_filter_lim (F : (T -> Prop) -> Prop) (x : T) :=
forall P, locally x P -> F P.
End Locally.

Lemma filterlim_const :
forall {T} {U : UniformSpace} {F : (T -> Prop) -> Prop} {FF : Filter F},
forall a : U, filterlim (fun _ => a) F (locally a).