--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+
+
+include "reals.ma".
+
+axiom continuo: (R → R) → Prop.
+axiom tends_to: (nat → R) → R → Prop.
+axiom tends_to_inf: (nat → R) → Prop.
+
+definition monotone_not_increasing ≝
+ λ alpha:nat→R.
+ ∀n:nat.alpha (S n) ≤ alpha n.
+
+definition inf_bounded ≝
+ λ alpha:nat → R.
+ ∃ m. ∀ n:nat. m ≤ alpha n.
+
+axiom converge: ∀ alpha.
+ monotone_not_increasing alpha →
+ inf_bounded alpha →
+ ∃ l. tends_to alpha l.
+
+definition punto_fisso :=
+ λ F:R→R. λ x. x = F x.
+
+let rec successione F x (n:nat) on n : R ≝
+ match n with
+ [ O ⇒ x
+ | S n ⇒ F (successione F x n)
+ ].
+
+axiom lim_punto_fisso:
+∀ F: R → R. ∀b:R. continuo F →
+ let alpha := successione F b in
+ ∀ l. tends_to alpha l →
+ punto_fisso F l.
+
+definition monotone_not_decreasing ≝
+ λ alpha:nat→R.
+ ∀n:nat.alpha n ≤ alpha (S n).
+
+definition sup_bounded ≝
+ λ alpha:nat → R.
+ ∃ m. ∀ n:nat. alpha n ≤ m.