Definition convert_approx_to_uniform_space:
nat -> uniform_space R_metric (fun _:point_set X => 0).
refine (fun n:nat => exist _ (proj1_sig (extension_approximation_seq n)) _).
destruct extension_approximation_seq as [g [? []]].
simpl.
unfold R_metric.
exists (1 - (2/3)^n).
red; intros.
destruct H.
rewrite H0; clear y H0.
destruct (a x).
unfold Rabs; destruct Rcase_abs; fourier.
Defined.

Lemma extension_approximation_seq_cauchy:
cauchy (uniform_metric R_metric (fun _:point_set X => 0)
R_metric_is_metric X_nonempty)
convert_approx_to_uniform_space.