Definition tearing_elem_fst (u : nat -> R) (v : R -> nat) (pen : enumeration R u v) (x : R)
  := exists n : nat, x = fst (proj1_sig (tearing_sequences u v pen n)).(* The limit of the first tearing sequence cannot be reached by u *)
Definition torn_number (u : nat -> R) (v : R -> nat) (pen : enumeration R u v) :
  {m : R | is_lub (tearing_elem_fst u v pen) m}.

Lemma first_indices_increasing (u : nat -> R) (v : R -> nat) (H : enumeration R u v) : forall n : nat, n <> 0 -> v (fst (proj1_sig (tearing_sequences u v H n))) < v (fst (proj1_sig (tearing_sequences u v H (S n)))).