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}.

(* Here is the contradiction : the torn number's index is above a sequence that tends to infinity *)
Lemma limit_index_above_all_indices (u : nat -> R) (v : R -> nat) (en : enumeration R u v) :
forall n : nat, v (fst (proj1_sig (tearing_sequences u v en (S n))))
< v (proj1_sig (torn_number u v en)).