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}.

(* The torn number is between both tearing sequences, so it could have been chosen at each step. *)
Lemma torn_number_below_second_sequence (u : nat -> R) (v : R -> nat)
(en : enumeration R u v) :
forall n : nat, Rlt (proj1_sig (torn_number u v en))
(snd (proj1_sig (tearing_sequences u v en n))).