proposition: Lemma Alembert_exp : Un_cv (fun n:nat => Rabs (/ INR (fact (S n)) * / / INR (fact n))) 0.
proof: Proof. unfold Un_cv; intros; destruct (Rgt_dec eps 1) as [Hgt|Hnotgt]. - split with 0%nat; intros; rewrite (simpl_fact n); unfold Rdist; rewrite (Rminus_0_r (Rabs (/ INR (S n)))); rewrite (Rabs_Rabsolu (/ INR (S n))); cut (/ INR (S n) > 0). + intro; rewrite (Rabs_pos_eq (/ INR (S n))). * cut (/ eps - 1 < 0). -- intro H2; generalize (Rlt_le_trans (/ eps - 1) 0 (INR n) H2 (pos_INR n)); clear H2; intro; unfold Rminus in H2; generalize (Rplus_lt_compat_l 1 (/ eps + -1) (INR n) H2); replace (1 + (/ eps + -1)) with (/ eps); [ clear H2; intro | ring ]. rewrite (Rplus_comm 1 (INR n)) in H2; rewrite <- (S_INR n) in H2; generalize (Rmult_gt_0_compat (/ INR (S n)) eps H1 H); intro; unfold Rgt in H3; generalize (Rmult_lt_compat_l (/ INR (S n) * eps) (/ eps) (INR (S n)) H3 H2); intro; rewrite (Rmult_assoc (/ INR (S n)) eps (/ eps)) in H4; rewrite (Rinv_r eps (Rlt_dichotomy_converse eps 0 (or_intror (eps < 0) H))) in H4; rewrite (let (H1, H2) := Rmult_ne (/ INR (S n)) in H1) in H4; rewrite (Rmult_comm (/ INR (S n))) in H4; rewrite (Rmult_assoc eps (/ INR (S n)) (INR (S n))) in H4; rewrite (Rinv_l (INR (S n)) (not_O_INR (S n) (not_eq_sym (O_S n)))) in H4; rewrite (let (H1, H2) := Rmult_ne eps in H1) in H4; assumption. -- apply Rlt_minus; unfold Rgt in Hgt; rewrite <- Rinv_1; apply (Rinv_lt_contravar 1 eps); auto; rewrite (let (H1, H2) := Rmult_ne eps in H2); unfold Rgt in H; assumption. * unfold Rgt in H1; apply Rlt_le; assumption. + unfold Rgt; apply Rinv_0_lt_compat; apply lt_INR_0; apply Nat.lt_0_succ. - cut (0 <= up (/ eps - 1))%Z. + intro; elim (IZN (up (/ eps - 1)) H0); intros; split with x; intros; rewrite (simpl_fact n); unfold Rdist; rewrite (Rminus_0_r (Rabs (/ INR (S n)))); rewrite (Rabs_Rabsolu (/ INR (S n))); cut (/ INR (S n) > 0). * intro; rewrite (Rabs_pos_eq (/ INR (S n))). -- cut (/ eps - 1 < INR x). ++ intro ; generalize (Rlt_le_trans (/ eps - 1) (INR x) (INR n) H4 (le_INR x n H2)); clear H4; intro; unfold Rminus in H4; generalize (Rplus_lt_compat_l 1 (/ eps + -1) (INR n) H4); replace (1 + (/ eps + -1)) with (/ eps); [ clear H4; intro | ring ]. rewrite (Rplus_comm 1 (INR n)) in H4; rewrite <- (S_INR n) in H4; generalize (Rmult_gt_0_compat (/ INR (S n)) eps H3 H); intro; unfold Rgt in H5; generalize (Rmult_lt_compat_l (/ INR (S n) * eps) (/ eps) (INR (S n)) H5 H4); intro; rewrite (Rmult_assoc (/ INR (S n)) eps (/ eps)) in H6; rewrite (Rinv_r eps (Rlt_dichotomy_converse eps 0 (or_intror (eps < 0) H))) in H6; rewrite (let (H1, H2) := Rmult_ne (/ INR (S n)) in H1) in H6; rewrite (Rmult_comm (/ INR (S n))) in H6; rewrite (Rmult_assoc eps (/ INR (S n)) (INR (S n))) in H6; rewrite (Rinv_l (INR (S n)) (not_O_INR (S n) (not_eq_sym (O_S n)))) in H6; rewrite (let (H1, H2) := Rmult_ne eps in H1) in H6; assumption. ++ cut (IZR (up (/ eps - 1)) = IZR (Z.of_nat x)); [ intro | rewrite H1; trivial ]. elim (archimed (/ eps - 1)); intros; clear H6; unfold Rgt in H5; rewrite H4 in H5; rewrite INR_IZR_INZ; assumption. -- unfold Rgt in H1; apply Rlt_le; assumption. * unfold Rgt; apply Rinv_0_lt_compat; apply lt_INR_0; apply Nat.lt_0_succ. + apply (le_O_IZR (up (/ eps - 1))); apply (Rle_trans 0 (/ eps - 1) (IZR (up (/ eps - 1)))). * generalize (Rnot_gt_le eps 1 Hnotgt); clear Hnotgt; unfold Rle; intro; elim H0; clear H0; intro. -- left; unfold Rgt in H; generalize (Rmult_lt_compat_l (/ eps) eps 1 (Rinv_0_lt_compat eps H) H0); rewrite (Rinv_l eps (not_eq_sym (Rlt_dichotomy_converse 0 eps (or_introl (0 > eps) H)))) ; rewrite (let (H1, H2) := Rmult_ne (/ eps) in H1); intro; fold (/ eps - 1 > 0); apply Rgt_minus; unfold Rgt; assumption. -- right; rewrite H0; rewrite Rinv_1; symmetry; apply Rminus_diag_eq; auto. * elim (archimed (/ eps - 1)); intros; clear H1; unfold Rgt in H0; apply Rlt_le; assumption. Qed.
imports: Rbase Rfunctions SeqSeries