Definition extension_approximation_seq: forall n:nat,
  { g0:point_set X -> point_set RTop |
    continuous g0 /\
    (forall x:point_set X, -1 + (2/3)^n <= g0 x <= 1 - (2/3)^n) /\
    (forall x:point_set (SubspaceTopology F),
      -(2/3)^n <= f x - g0 (subspace_inc F x) <= (2/3)^n) }.

(* Specifications: *)
Lemma extension_approximation_seq_diff: forall (n:nat) (x:point_set X), -(1/3 * (2/3)^n) <= proj1_sig (extension_approximation_seq (S n)) x - proj1_sig (extension_approximation_seq n) x <= 1/3*(2/3)^n.
Lemma Rle_R1_pow: forall (x:R) (m n:nat), 0 <= x <= 1 -> (m <= n)%nat -> x^n <= x^m.
Lemma extension_approximation_seq_cauchy_aux: forall (m n:nat) (x:point_set X), Rabs (proj1_sig (extension_approximation_seq m) x - proj1_sig (extension_approximation_seq n) x) <= Rabs ((2/3)^m - (2/3)^n).