Definition ball_norm (x : V) (eps : R) (y : V) := norm (minus y x) < eps.
Definition locally_norm (x : V) (P : V -> Prop) :=
exists eps : posreal, forall y, ball_norm x eps y -> P y.

Lemma is_filter_lim_unique :
forall {F} {FF : ProperFilter' F} (x y : V),
is_filter_lim F x -> is_filter_lim F y -> x = y.