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Diffstat (limited to 'theories/Reals/SeqProp.v')
| -rw-r--r-- | theories/Reals/SeqProp.v | 1295 |
1 files changed, 1295 insertions, 0 deletions
diff --git a/theories/Reals/SeqProp.v b/theories/Reals/SeqProp.v new file mode 100644 index 00000000..34f9fd72 --- /dev/null +++ b/theories/Reals/SeqProp.v @@ -0,0 +1,1295 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: SeqProp.v,v 1.13.2.1 2004/07/16 19:31:15 herbelin Exp $ i*) + +Require Import Rbase. +Require Import Rfunctions. +Require Import Rseries. +Require Import Classical. +Require Import Max. +Open Local Scope R_scope. + +Definition Un_decreasing (Un:nat -> R) : Prop := + forall n:nat, Un (S n) <= Un n. +Definition opp_seq (Un:nat -> R) (n:nat) : R := - Un n. +Definition has_ub (Un:nat -> R) : Prop := bound (EUn Un). +Definition has_lb (Un:nat -> R) : Prop := bound (EUn (opp_seq Un)). + +(**********) +Lemma growing_cv : + forall Un:nat -> R, + Un_growing Un -> has_ub Un -> sigT (fun l:R => Un_cv Un l). +unfold Un_growing, Un_cv in |- *; intros; + destruct (completeness (EUn Un) H0 (EUn_noempty Un)) as [x [H2 H3]]. + exists x; intros eps H1. + unfold is_upper_bound in H2, H3. +assert (H5 : forall n:nat, Un n <= x). + intro n; apply (H2 (Un n) (Un_in_EUn Un n)). +cut (exists N : nat, x - eps < Un N). +intro H6; destruct H6 as [N H6]; exists N. +intros n H7; unfold R_dist in |- *; apply (Rabs_def1 (Un n - x) eps). +unfold Rgt in H1. + apply (Rle_lt_trans (Un n - x) 0 eps (Rle_minus (Un n) x (H5 n)) H1). +fold Un_growing in H; generalize (growing_prop Un n N H H7); intro H8. + generalize + (Rlt_le_trans (x - eps) (Un N) (Un n) H6 (Rge_le (Un n) (Un N) H8)); + intro H9; generalize (Rplus_lt_compat_l (- x) (x - eps) (Un n) H9); + unfold Rminus in |- *; rewrite <- (Rplus_assoc (- x) x (- eps)); + rewrite (Rplus_comm (- x) (Un n)); fold (Un n - x) in |- *; + rewrite Rplus_opp_l; rewrite (let (H1, H2) := Rplus_ne (- eps) in H2); + trivial. +cut (~ (forall N:nat, Un N <= x - eps)). +intro H6; apply (not_all_not_ex nat (fun N:nat => x - eps < Un N)). + intro H7; apply H6; intro N; apply Rnot_lt_le; apply H7. +intro H7; generalize (Un_bound_imp Un (x - eps) H7); intro H8; + unfold is_upper_bound in H8; generalize (H3 (x - eps) H8); + apply Rlt_not_le; apply tech_Rgt_minus; exact H1. +Qed. + +Lemma decreasing_growing : + forall Un:nat -> R, Un_decreasing Un -> Un_growing (opp_seq Un). +intro. +unfold Un_growing, opp_seq, Un_decreasing in |- *. +intros. +apply Ropp_le_contravar. +apply H. +Qed. + +Lemma decreasing_cv : + forall Un:nat -> R, + Un_decreasing Un -> has_lb Un -> sigT (fun l:R => Un_cv Un l). +intros. +cut (sigT (fun l:R => Un_cv (opp_seq Un) l) -> sigT (fun l:R => Un_cv Un l)). +intro. +apply X. +apply growing_cv. +apply decreasing_growing; assumption. +exact H0. +intro. +elim X; intros. +apply existT with (- x). +unfold Un_cv in p. +unfold R_dist in p. +unfold opp_seq in p. +unfold Un_cv in |- *. +unfold R_dist in |- *. +intros. +elim (p eps H1); intros. +exists x0; intros. +assert (H4 := H2 n H3). +rewrite <- Rabs_Ropp. +replace (- (Un n - - x)) with (- Un n - x); [ assumption | ring ]. +Qed. + +(***********) +Lemma maj_sup : + forall Un:nat -> R, has_ub Un -> sigT (fun l:R => is_lub (EUn Un) l). +intros. +unfold has_ub in H. +apply completeness. +assumption. +exists (Un 0%nat). +unfold EUn in |- *. +exists 0%nat; reflexivity. +Qed. + +(**********) +Lemma min_inf : + forall Un:nat -> R, + has_lb Un -> sigT (fun l:R => is_lub (EUn (opp_seq Un)) l). +intros; unfold has_lb in H. +apply completeness. +assumption. +exists (- Un 0%nat). +exists 0%nat. +reflexivity. +Qed. + +Definition majorant (Un:nat -> R) (pr:has_ub Un) : R := + match maj_sup Un pr with + | existT a b => a + end. + +Definition minorant (Un:nat -> R) (pr:has_lb Un) : R := + match min_inf Un pr with + | existT a b => - a + end. + +Lemma maj_ss : + forall (Un:nat -> R) (k:nat), + has_ub Un -> has_ub (fun i:nat => Un (k + i)%nat). +intros. +unfold has_ub in H. +unfold bound in H. +elim H; intros. +unfold is_upper_bound in H0. +unfold has_ub in |- *. +exists x. +unfold is_upper_bound in |- *. +intros. +apply H0. +elim H1; intros. +exists (k + x1)%nat; assumption. +Qed. + +Lemma min_ss : + forall (Un:nat -> R) (k:nat), + has_lb Un -> has_lb (fun i:nat => Un (k + i)%nat). +intros. +unfold has_lb in H. +unfold bound in H. +elim H; intros. +unfold is_upper_bound in H0. +unfold has_lb in |- *. +exists x. +unfold is_upper_bound in |- *. +intros. +apply H0. +elim H1; intros. +exists (k + x1)%nat; assumption. +Qed. + +Definition sequence_majorant (Un:nat -> R) (pr:has_ub Un) + (i:nat) : R := majorant (fun k:nat => Un (i + k)%nat) (maj_ss Un i pr). + +Definition sequence_minorant (Un:nat -> R) (pr:has_lb Un) + (i:nat) : R := minorant (fun k:nat => Un (i + k)%nat) (min_ss Un i pr). + +Lemma Wn_decreasing : + forall (Un:nat -> R) (pr:has_ub Un), Un_decreasing (sequence_majorant Un pr). +intros. +unfold Un_decreasing in |- *. +intro. +unfold sequence_majorant in |- *. +assert (H := maj_sup (fun k:nat => Un (S n + k)%nat) (maj_ss Un (S n) pr)). +assert (H0 := maj_sup (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr)). +elim H; intros. +elim H0; intros. +cut (majorant (fun k:nat => Un (S n + k)%nat) (maj_ss Un (S n) pr) = x); + [ intro Maj1; rewrite Maj1 | idtac ]. +cut (majorant (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr) = x0); + [ intro Maj2; rewrite Maj2 | idtac ]. +unfold is_lub in p. +unfold is_lub in p0. +elim p; intros. +apply H2. +elim p0; intros. +unfold is_upper_bound in |- *. +intros. +unfold is_upper_bound in H3. +apply H3. +elim H5; intros. +exists (1 + x2)%nat. +replace (n + (1 + x2))%nat with (S n + x2)%nat. +assumption. +replace (S n) with (1 + n)%nat; [ ring | ring ]. +cut + (is_lub (EUn (fun k:nat => Un (n + k)%nat)) + (majorant (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr))). +intro. +unfold is_lub in p0; unfold is_lub in H1. +elim p0; intros; elim H1; intros. +assert (H6 := H5 x0 H2). +assert + (H7 := H3 (majorant (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr)) H4). +apply Rle_antisym; assumption. +unfold majorant in |- *. +case (maj_sup (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr)). +trivial. +cut + (is_lub (EUn (fun k:nat => Un (S n + k)%nat)) + (majorant (fun k:nat => Un (S n + k)%nat) (maj_ss Un (S n) pr))). +intro. +unfold is_lub in p; unfold is_lub in H1. +elim p; intros; elim H1; intros. +assert (H6 := H5 x H2). +assert + (H7 := + H3 (majorant (fun k:nat => Un (S n + k)%nat) (maj_ss Un (S n) pr)) H4). +apply Rle_antisym; assumption. +unfold majorant in |- *. +case (maj_sup (fun k:nat => Un (S n + k)%nat) (maj_ss Un (S n) pr)). +trivial. +Qed. + +Lemma Vn_growing : + forall (Un:nat -> R) (pr:has_lb Un), Un_growing (sequence_minorant Un pr). +intros. +unfold Un_growing in |- *. +intro. +unfold sequence_minorant in |- *. +assert (H := min_inf (fun k:nat => Un (S n + k)%nat) (min_ss Un (S n) pr)). +assert (H0 := min_inf (fun k:nat => Un (n + k)%nat) (min_ss Un n pr)). +elim H; intros. +elim H0; intros. +cut (minorant (fun k:nat => Un (S n + k)%nat) (min_ss Un (S n) pr) = - x); + [ intro Maj1; rewrite Maj1 | idtac ]. +cut (minorant (fun k:nat => Un (n + k)%nat) (min_ss Un n pr) = - x0); + [ intro Maj2; rewrite Maj2 | idtac ]. +unfold is_lub in p. +unfold is_lub in p0. +elim p; intros. +apply Ropp_le_contravar. +apply H2. +elim p0; intros. +unfold is_upper_bound in |- *. +intros. +unfold is_upper_bound in H3. +apply H3. +elim H5; intros. +exists (1 + x2)%nat. +unfold opp_seq in H6. +unfold opp_seq in |- *. +replace (n + (1 + x2))%nat with (S n + x2)%nat. +assumption. +replace (S n) with (1 + n)%nat; [ ring | ring ]. +cut + (is_lub (EUn (opp_seq (fun k:nat => Un (n + k)%nat))) + (- minorant (fun k:nat => Un (n + k)%nat) (min_ss Un n pr))). +intro. +unfold is_lub in p0; unfold is_lub in H1. +elim p0; intros; elim H1; intros. +assert (H6 := H5 x0 H2). +assert + (H7 := H3 (- minorant (fun k:nat => Un (n + k)%nat) (min_ss Un n pr)) H4). +rewrite <- + (Ropp_involutive (minorant (fun k:nat => Un (n + k)%nat) (min_ss Un n pr))) + . +apply Ropp_eq_compat; apply Rle_antisym; assumption. +unfold minorant in |- *. +case (min_inf (fun k:nat => Un (n + k)%nat) (min_ss Un n pr)). +intro; rewrite Ropp_involutive. +trivial. +cut + (is_lub (EUn (opp_seq (fun k:nat => Un (S n + k)%nat))) + (- minorant (fun k:nat => Un (S n + k)%nat) (min_ss Un (S n) pr))). +intro. +unfold is_lub in p; unfold is_lub in H1. +elim p; intros; elim H1; intros. +assert (H6 := H5 x H2). +assert + (H7 := + H3 (- minorant (fun k:nat => Un (S n + k)%nat) (min_ss Un (S n) pr)) H4). +rewrite <- + (Ropp_involutive + (minorant (fun k:nat => Un (S n + k)%nat) (min_ss Un (S n) pr))) + . +apply Ropp_eq_compat; apply Rle_antisym; assumption. +unfold minorant in |- *. +case (min_inf (fun k:nat => Un (S n + k)%nat) (min_ss Un (S n) pr)). +intro; rewrite Ropp_involutive. +trivial. +Qed. + +(**********) +Lemma Vn_Un_Wn_order : + forall (Un:nat -> R) (pr1:has_ub Un) (pr2:has_lb Un) + (n:nat), sequence_minorant Un pr2 n <= Un n <= sequence_majorant Un pr1 n. +intros. +split. +unfold sequence_minorant in |- *. +cut + (sigT (fun l:R => is_lub (EUn (opp_seq (fun i:nat => Un (n + i)%nat))) l)). +intro. +elim X; intros. +replace (minorant (fun k:nat => Un (n + k)%nat) (min_ss Un n pr2)) with (- x). +unfold is_lub in p. +elim p; intros. +unfold is_upper_bound in H. +rewrite <- (Ropp_involutive (Un n)). +apply Ropp_le_contravar. +apply H. +exists 0%nat. +unfold opp_seq in |- *. +replace (n + 0)%nat with n; [ reflexivity | ring ]. +cut + (is_lub (EUn (opp_seq (fun k:nat => Un (n + k)%nat))) + (- minorant (fun k:nat => Un (n + k)%nat) (min_ss Un n pr2))). +intro. +unfold is_lub in p; unfold is_lub in H. +elim p; intros; elim H; intros. +assert (H4 := H3 x H0). +assert + (H5 := H1 (- minorant (fun k:nat => Un (n + k)%nat) (min_ss Un n pr2)) H2). +rewrite <- + (Ropp_involutive (minorant (fun k:nat => Un (n + k)%nat) (min_ss Un n pr2))) + . +apply Ropp_eq_compat; apply Rle_antisym; assumption. +unfold minorant in |- *. +case (min_inf (fun k:nat => Un (n + k)%nat) (min_ss Un n pr2)). +intro; rewrite Ropp_involutive. +trivial. +apply min_inf. +apply min_ss; assumption. +unfold sequence_majorant in |- *. +cut (sigT (fun l:R => is_lub (EUn (fun i:nat => Un (n + i)%nat)) l)). +intro. +elim X; intros. +replace (majorant (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr1)) with x. +unfold is_lub in p. +elim p; intros. +unfold is_upper_bound in H. +apply H. +exists 0%nat. +replace (n + 0)%nat with n; [ reflexivity | ring ]. +cut + (is_lub (EUn (fun k:nat => Un (n + k)%nat)) + (majorant (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr1))). +intro. +unfold is_lub in p; unfold is_lub in H. +elim p; intros; elim H; intros. +assert (H4 := H3 x H0). +assert + (H5 := H1 (majorant (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr1)) H2). +apply Rle_antisym; assumption. +unfold majorant in |- *. +case (maj_sup (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr1)). +intro; trivial. +apply maj_sup. +apply maj_ss; assumption. +Qed. + +Lemma min_maj : + forall (Un:nat -> R) (pr1:has_ub Un) (pr2:has_lb Un), + has_ub (sequence_minorant Un pr2). +intros. +assert (H := Vn_Un_Wn_order Un pr1 pr2). +unfold has_ub in |- *. +unfold bound in |- *. +unfold has_ub in pr1. +unfold bound in pr1. +elim pr1; intros. +exists x. +unfold is_upper_bound in |- *. +intros. +unfold is_upper_bound in H0. +elim H1; intros. +rewrite H2. +apply Rle_trans with (Un x1). +assert (H3 := H x1); elim H3; intros; assumption. +apply H0. +exists x1; reflexivity. +Qed. + +Lemma maj_min : + forall (Un:nat -> R) (pr1:has_ub Un) (pr2:has_lb Un), + has_lb (sequence_majorant Un pr1). +intros. +assert (H := Vn_Un_Wn_order Un pr1 pr2). +unfold has_lb in |- *. +unfold bound in |- *. +unfold has_lb in pr2. +unfold bound in pr2. +elim pr2; intros. +exists x. +unfold is_upper_bound in |- *. +intros. +unfold is_upper_bound in H0. +elim H1; intros. +rewrite H2. +apply Rle_trans with (opp_seq Un x1). +assert (H3 := H x1); elim H3; intros. +unfold opp_seq in |- *; apply Ropp_le_contravar. +assumption. +apply H0. +exists x1; reflexivity. +Qed. + +(**********) +Lemma cauchy_maj : forall Un:nat -> R, Cauchy_crit Un -> has_ub Un. +intros. +unfold has_ub in |- *. +apply cauchy_bound. +assumption. +Qed. + +(**********) +Lemma cauchy_opp : + forall Un:nat -> R, Cauchy_crit Un -> Cauchy_crit (opp_seq Un). +intro. +unfold Cauchy_crit in |- *. +unfold R_dist in |- *. +intros. +elim (H eps H0); intros. +exists x; intros. +unfold opp_seq in |- *. +rewrite <- Rabs_Ropp. +replace (- (- Un n - - Un m)) with (Un n - Un m); + [ apply H1; assumption | ring ]. +Qed. + +(**********) +Lemma cauchy_min : forall Un:nat -> R, Cauchy_crit Un -> has_lb Un. +intros. +unfold has_lb in |- *. +assert (H0 := cauchy_opp _ H). +apply cauchy_bound. +assumption. +Qed. + +(**********) +Lemma maj_cv : + forall (Un:nat -> R) (pr:Cauchy_crit Un), + sigT (fun l:R => Un_cv (sequence_majorant Un (cauchy_maj Un pr)) l). +intros. +apply decreasing_cv. +apply Wn_decreasing. +apply maj_min. +apply cauchy_min. +assumption. +Qed. + +(**********) +Lemma min_cv : + forall (Un:nat -> R) (pr:Cauchy_crit Un), + sigT (fun l:R => Un_cv (sequence_minorant Un (cauchy_min Un pr)) l). +intros. +apply growing_cv. +apply Vn_growing. +apply min_maj. +apply cauchy_maj. +assumption. +Qed. + +Lemma cond_eq : + forall x y:R, (forall eps:R, 0 < eps -> Rabs (x - y) < eps) -> x = y. +intros. +case (total_order_T x y); intro. +elim s; intro. +cut (0 < y - x). +intro. +assert (H1 := H (y - x) H0). +rewrite <- Rabs_Ropp in H1. +cut (- (x - y) = y - x); [ intro; rewrite H2 in H1 | ring ]. +rewrite Rabs_right in H1. +elim (Rlt_irrefl _ H1). +left; assumption. +apply Rplus_lt_reg_r with x. +rewrite Rplus_0_r; replace (x + (y - x)) with y; [ assumption | ring ]. +assumption. +cut (0 < x - y). +intro. +assert (H1 := H (x - y) H0). +rewrite Rabs_right in H1. +elim (Rlt_irrefl _ H1). +left; assumption. +apply Rplus_lt_reg_r with y. +rewrite Rplus_0_r; replace (y + (x - y)) with x; [ assumption | ring ]. +Qed. + +Lemma not_Rlt : forall r1 r2:R, ~ r1 < r2 -> r1 >= r2. +intros r1 r2; generalize (Rtotal_order r1 r2); unfold Rge in |- *. +tauto. +Qed. + +(**********) +Lemma approx_maj : + forall (Un:nat -> R) (pr:has_ub Un) (eps:R), + 0 < eps -> exists k : nat, Rabs (majorant Un pr - Un k) < eps. +intros. +set (P := fun k:nat => Rabs (majorant Un pr - Un k) < eps). +unfold P in |- *. +cut + ((exists k : nat, P k) -> + exists k : nat, Rabs (majorant Un pr - Un k) < eps). +intros. +apply H0. +apply not_all_not_ex. +red in |- *; intro. +2: unfold P in |- *; trivial. +unfold P in H1. +cut (forall n:nat, Rabs (majorant Un pr - Un n) >= eps). +intro. +cut (is_lub (EUn Un) (majorant Un pr)). +intro. +unfold is_lub in H3. +unfold is_upper_bound in H3. +elim H3; intros. +cut (forall n:nat, eps <= majorant Un pr - Un n). +intro. +cut (forall n:nat, Un n <= majorant Un pr - eps). +intro. +cut (forall x:R, EUn Un x -> x <= majorant Un pr - eps). +intro. +assert (H9 := H5 (majorant Un pr - eps) H8). +cut (eps <= 0). +intro. +elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ H H10)). +apply Rplus_le_reg_l with (majorant Un pr - eps). +rewrite Rplus_0_r. +replace (majorant Un pr - eps + eps) with (majorant Un pr); + [ assumption | ring ]. +intros. +unfold EUn in H8. +elim H8; intros. +rewrite H9; apply H7. +intro. +assert (H7 := H6 n). +apply Rplus_le_reg_l with (eps - Un n). +replace (eps - Un n + Un n) with eps. +replace (eps - Un n + (majorant Un pr - eps)) with (majorant Un pr - Un n). +assumption. +ring. +ring. +intro. +assert (H6 := H2 n). +rewrite Rabs_right in H6. +apply Rge_le. +assumption. +apply Rle_ge. +apply Rplus_le_reg_l with (Un n). +rewrite Rplus_0_r; + replace (Un n + (majorant Un pr - Un n)) with (majorant Un pr); + [ apply H4 | ring ]. +exists n; reflexivity. +unfold majorant in |- *. +case (maj_sup Un pr). +trivial. +intro. +assert (H2 := H1 n). +apply not_Rlt; assumption. +Qed. + +(**********) +Lemma approx_min : + forall (Un:nat -> R) (pr:has_lb Un) (eps:R), + 0 < eps -> exists k : nat, Rabs (minorant Un pr - Un k) < eps. +intros. +set (P := fun k:nat => Rabs (minorant Un pr - Un k) < eps). +unfold P in |- *. +cut + ((exists k : nat, P k) -> + exists k : nat, Rabs (minorant Un pr - Un k) < eps). +intros. +apply H0. +apply not_all_not_ex. +red in |- *; intro. +2: unfold P in |- *; trivial. +unfold P in H1. +cut (forall n:nat, Rabs (minorant Un pr - Un n) >= eps). +intro. +cut (is_lub (EUn (opp_seq Un)) (- minorant Un pr)). +intro. +unfold is_lub in H3. +unfold is_upper_bound in H3. +elim H3; intros. +cut (forall n:nat, eps <= Un n - minorant Un pr). +intro. +cut (forall n:nat, opp_seq Un n <= - minorant Un pr - eps). +intro. +cut (forall x:R, EUn (opp_seq Un) x -> x <= - minorant Un pr - eps). +intro. +assert (H9 := H5 (- minorant Un pr - eps) H8). +cut (eps <= 0). +intro. +elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ H H10)). +apply Rplus_le_reg_l with (- minorant Un pr - eps). +rewrite Rplus_0_r. +replace (- minorant Un pr - eps + eps) with (- minorant Un pr); + [ assumption | ring ]. +intros. +unfold EUn in H8. +elim H8; intros. +rewrite H9; apply H7. +intro. +assert (H7 := H6 n). +unfold opp_seq in |- *. +apply Rplus_le_reg_l with (eps + Un n). +replace (eps + Un n + - Un n) with eps. +replace (eps + Un n + (- minorant Un pr - eps)) with (Un n - minorant Un pr). +assumption. +ring. +ring. +intro. +assert (H6 := H2 n). +rewrite Rabs_left1 in H6. +apply Rge_le. +replace (Un n - minorant Un pr) with (- (minorant Un pr - Un n)); + [ assumption | ring ]. +apply Rplus_le_reg_l with (- minorant Un pr). +rewrite Rplus_0_r; + replace (- minorant Un pr + (minorant Un pr - Un n)) with (- Un n). +apply H4. +exists n; reflexivity. +ring. +unfold minorant in |- *. +case (min_inf Un pr). +intro. +rewrite Ropp_involutive. +trivial. +intro. +assert (H2 := H1 n). +apply not_Rlt; assumption. +Qed. + +(* Unicity of limit for convergent sequences *) +Lemma UL_sequence : + forall (Un:nat -> R) (l1 l2:R), Un_cv Un l1 -> Un_cv Un l2 -> l1 = l2. +intros Un l1 l2; unfold Un_cv in |- *; unfold R_dist in |- *; intros. +apply cond_eq. +intros; cut (0 < eps / 2); + [ intro + | unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ]. +elim (H (eps / 2) H2); intros. +elim (H0 (eps / 2) H2); intros. +set (N := max x x0). +apply Rle_lt_trans with (Rabs (l1 - Un N) + Rabs (Un N - l2)). +replace (l1 - l2) with (l1 - Un N + (Un N - l2)); + [ apply Rabs_triang | ring ]. +rewrite (double_var eps); apply Rplus_lt_compat. +rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; apply H3; + unfold ge, N in |- *; apply le_max_l. +apply H4; unfold ge, N in |- *; apply le_max_r. +Qed. + +(**********) +Lemma CV_plus : + forall (An Bn:nat -> R) (l1 l2:R), + Un_cv An l1 -> Un_cv Bn l2 -> Un_cv (fun i:nat => An i + Bn i) (l1 + l2). +unfold Un_cv in |- *; unfold R_dist in |- *; intros. +cut (0 < eps / 2); + [ intro + | unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ]. +elim (H (eps / 2) H2); intros. +elim (H0 (eps / 2) H2); intros. +set (N := max x x0). +exists N; intros. +replace (An n + Bn n - (l1 + l2)) with (An n - l1 + (Bn n - l2)); + [ idtac | ring ]. +apply Rle_lt_trans with (Rabs (An n - l1) + Rabs (Bn n - l2)). +apply Rabs_triang. +rewrite (double_var eps); apply Rplus_lt_compat. +apply H3; unfold ge in |- *; apply le_trans with N; + [ unfold N in |- *; apply le_max_l | assumption ]. +apply H4; unfold ge in |- *; apply le_trans with N; + [ unfold N in |- *; apply le_max_r | assumption ]. +Qed. + +(**********) +Lemma cv_cvabs : + forall (Un:nat -> R) (l:R), + Un_cv Un l -> Un_cv (fun i:nat => Rabs (Un i)) (Rabs l). +unfold Un_cv in |- *; unfold R_dist in |- *; intros. +elim (H eps H0); intros. +exists x; intros. +apply Rle_lt_trans with (Rabs (Un n - l)). +apply Rabs_triang_inv2. +apply H1; assumption. +Qed. + +(**********) +Lemma CV_Cauchy : + forall Un:nat -> R, sigT (fun l:R => Un_cv Un l) -> Cauchy_crit Un. +intros; elim X; intros. +unfold Cauchy_crit in |- *; intros. +unfold Un_cv in p; unfold R_dist in p. +cut (0 < eps / 2); + [ intro + | unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ]. +elim (p (eps / 2) H0); intros. +exists x0; intros. +unfold R_dist in |- *; + apply Rle_lt_trans with (Rabs (Un n - x) + Rabs (x - Un m)). +replace (Un n - Un m) with (Un n - x + (x - Un m)); + [ apply Rabs_triang | ring ]. +rewrite (double_var eps); apply Rplus_lt_compat. +apply H1; assumption. +rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; apply H1; assumption. +Qed. + +(**********) +Lemma maj_by_pos : + forall Un:nat -> R, + sigT (fun l:R => Un_cv Un l) -> + exists l : R, 0 < l /\ (forall n:nat, Rabs (Un n) <= l). +intros; elim X; intros. +cut (sigT (fun l:R => Un_cv (fun k:nat => Rabs (Un k)) l)). +intro. +assert (H := CV_Cauchy (fun k:nat => Rabs (Un k)) X0). +assert (H0 := cauchy_bound (fun k:nat => Rabs (Un k)) H). +elim H0; intros. +exists (x0 + 1). +cut (0 <= x0). +intro. +split. +apply Rplus_le_lt_0_compat; [ assumption | apply Rlt_0_1 ]. +intros. +apply Rle_trans with x0. +unfold is_upper_bound in H1. +apply H1. +exists n; reflexivity. +pattern x0 at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left; + apply Rlt_0_1. +apply Rle_trans with (Rabs (Un 0%nat)). +apply Rabs_pos. +unfold is_upper_bound in H1. +apply H1. +exists 0%nat; reflexivity. +apply existT with (Rabs x). +apply cv_cvabs; assumption. +Qed. + +(**********) +Lemma CV_mult : + forall (An Bn:nat -> R) (l1 l2:R), + Un_cv An l1 -> Un_cv Bn l2 -> Un_cv (fun i:nat => An i * Bn i) (l1 * l2). +intros. +cut (sigT (fun l:R => Un_cv An l)). +intro. +assert (H1 := maj_by_pos An X). +elim H1; intros M H2. +elim H2; intros. +unfold Un_cv in |- *; unfold R_dist in |- *; intros. +cut (0 < eps / (2 * M)). +intro. +case (Req_dec l2 0); intro. +unfold Un_cv in H0; unfold R_dist in H0. +elim (H0 (eps / (2 * M)) H6); intros. +exists x; intros. +apply Rle_lt_trans with + (Rabs (An n * Bn n - An n * l2) + Rabs (An n * l2 - l1 * l2)). +replace (An n * Bn n - l1 * l2) with + (An n * Bn n - An n * l2 + (An n * l2 - l1 * l2)); + [ apply Rabs_triang | ring ]. +replace (Rabs (An n * Bn n - An n * l2)) with + (Rabs (An n) * Rabs (Bn n - l2)). +replace (Rabs (An n * l2 - l1 * l2)) with 0. +rewrite Rplus_0_r. +apply Rle_lt_trans with (M * Rabs (Bn n - l2)). +do 2 rewrite <- (Rmult_comm (Rabs (Bn n - l2))). +apply Rmult_le_compat_l. +apply Rabs_pos. +apply H4. +apply Rmult_lt_reg_l with (/ M). +apply Rinv_0_lt_compat; apply H3. +rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_l; rewrite (Rmult_comm (/ M)). +apply Rlt_trans with (eps / (2 * M)). +apply H8; assumption. +unfold Rdiv in |- *; rewrite Rinv_mult_distr. +apply Rmult_lt_reg_l with 2. +prove_sup0. +replace (2 * (eps * (/ 2 * / M))) with (2 * / 2 * (eps * / M)); + [ idtac | ring ]. +rewrite <- Rinv_r_sym. +rewrite Rmult_1_l; rewrite double. +pattern (eps * / M) at 1 in |- *; rewrite <- Rplus_0_r. +apply Rplus_lt_compat_l; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; assumption ]. +discrR. +discrR. +red in |- *; intro; rewrite H10 in H3; elim (Rlt_irrefl _ H3). +red in |- *; intro; rewrite H10 in H3; elim (Rlt_irrefl _ H3). +rewrite H7; do 2 rewrite Rmult_0_r; unfold Rminus in |- *; + rewrite Rplus_opp_r; rewrite Rabs_R0; reflexivity. +replace (An n * Bn n - An n * l2) with (An n * (Bn n - l2)); [ idtac | ring ]. +symmetry in |- *; apply Rabs_mult. +cut (0 < eps / (2 * Rabs l2)). +intro. +unfold Un_cv in H; unfold R_dist in H; unfold Un_cv in H0; + unfold R_dist in H0. +elim (H (eps / (2 * Rabs l2)) H8); intros N1 H9. +elim (H0 (eps / (2 * M)) H6); intros N2 H10. +set (N := max N1 N2). +exists N; intros. +apply Rle_lt_trans with + (Rabs (An n * Bn n - An n * l2) + Rabs (An n * l2 - l1 * l2)). +replace (An n * Bn n - l1 * l2) with + (An n * Bn n - An n * l2 + (An n * l2 - l1 * l2)); + [ apply Rabs_triang | ring ]. +replace (Rabs (An n * Bn n - An n * l2)) with + (Rabs (An n) * Rabs (Bn n - l2)). +replace (Rabs (An n * l2 - l1 * l2)) with (Rabs l2 * Rabs (An n - l1)). +rewrite (double_var eps); apply Rplus_lt_compat. +apply Rle_lt_trans with (M * Rabs (Bn n - l2)). +do 2 rewrite <- (Rmult_comm (Rabs (Bn n - l2))). +apply Rmult_le_compat_l. +apply Rabs_pos. +apply H4. +apply Rmult_lt_reg_l with (/ M). +apply Rinv_0_lt_compat; apply H3. +rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_l; rewrite (Rmult_comm (/ M)). +apply Rlt_le_trans with (eps / (2 * M)). +apply H10. +unfold ge in |- *; apply le_trans with N. +unfold N in |- *; apply le_max_r. +assumption. +unfold Rdiv in |- *; rewrite Rinv_mult_distr. +right; ring. +discrR. +red in |- *; intro; rewrite H12 in H3; elim (Rlt_irrefl _ H3). +red in |- *; intro; rewrite H12 in H3; elim (Rlt_irrefl _ H3). +apply Rmult_lt_reg_l with (/ Rabs l2). +apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption. +rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_l; apply Rlt_le_trans with (eps / (2 * Rabs l2)). +apply H9. +unfold ge in |- *; apply le_trans with N. +unfold N in |- *; apply le_max_l. +assumption. +unfold Rdiv in |- *; right; rewrite Rinv_mult_distr. +ring. +discrR. +apply Rabs_no_R0; assumption. +apply Rabs_no_R0; assumption. +replace (An n * l2 - l1 * l2) with (l2 * (An n - l1)); + [ symmetry in |- *; apply Rabs_mult | ring ]. +replace (An n * Bn n - An n * l2) with (An n * (Bn n - l2)); + [ symmetry in |- *; apply Rabs_mult | ring ]. +unfold Rdiv in |- *; apply Rmult_lt_0_compat. +assumption. +apply Rinv_0_lt_compat; apply Rmult_lt_0_compat; + [ prove_sup0 | apply Rabs_pos_lt; assumption ]. +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption + | apply Rinv_0_lt_compat; apply Rmult_lt_0_compat; + [ prove_sup0 | assumption ] ]. +apply existT with l1; assumption. +Qed. + +Lemma tech9 : + forall Un:nat -> R, + Un_growing Un -> forall m n:nat, (m <= n)%nat -> Un m <= Un n. +intros; unfold Un_growing in H. +induction n as [| n Hrecn]. +induction m as [| m Hrecm]. +right; reflexivity. +elim (le_Sn_O _ H0). +cut ((m <= n)%nat \/ m = S n). +intro; elim H1; intro. +apply Rle_trans with (Un n). +apply Hrecn; assumption. +apply H. +rewrite H2; right; reflexivity. +inversion H0. +right; reflexivity. +left; assumption. +Qed. + +Lemma tech10 : + forall (Un:nat -> R) (x:R), Un_growing Un -> is_lub (EUn Un) x -> Un_cv Un x. +intros; cut (bound (EUn Un)). +intro; assert (H2 := Un_cv_crit _ H H1). +elim H2; intros. +case (total_order_T x x0); intro. +elim s; intro. +cut (forall n:nat, Un n <= x). +intro; unfold Un_cv in H3; cut (0 < x0 - x). +intro; elim (H3 (x0 - x) H5); intros. +cut (x1 >= x1)%nat. +intro; assert (H8 := H6 x1 H7). +unfold R_dist in H8; rewrite Rabs_left1 in H8. +rewrite Ropp_minus_distr in H8; unfold Rminus in H8. +assert (H9 := Rplus_lt_reg_r x0 _ _ H8). +assert (H10 := Ropp_lt_cancel _ _ H9). +assert (H11 := H4 x1). +elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ H10 H11)). +apply Rle_minus; apply Rle_trans with x. +apply H4. +left; assumption. +unfold ge in |- *; apply le_n. +apply Rgt_minus; assumption. +intro; unfold is_lub in H0; unfold is_upper_bound in H0; elim H0; intros. +apply H4; unfold EUn in |- *; exists n; reflexivity. +rewrite b; assumption. +cut (forall n:nat, Un n <= x0). +intro; unfold is_lub in H0; unfold is_upper_bound in H0; elim H0; intros. +cut (forall y:R, EUn Un y -> y <= x0). +intro; assert (H8 := H6 _ H7). +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H8 r)). +unfold EUn in |- *; intros; elim H7; intros. +rewrite H8; apply H4. +intro; case (Rle_dec (Un n) x0); intro. +assumption. +cut (forall n0:nat, (n <= n0)%nat -> x0 < Un n0). +intro; unfold Un_cv in H3; cut (0 < Un n - x0). +intro; elim (H3 (Un n - x0) H5); intros. +cut (max n x1 >= x1)%nat. +intro; assert (H8 := H6 (max n x1) H7). +unfold R_dist in H8. +rewrite Rabs_right in H8. +unfold Rminus in H8; do 2 rewrite <- (Rplus_comm (- x0)) in H8. +assert (H9 := Rplus_lt_reg_r _ _ _ H8). +cut (Un n <= Un (max n x1)). +intro; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H10 H9)). +apply tech9; [ assumption | apply le_max_l ]. +apply Rge_trans with (Un n - x0). +unfold Rminus in |- *; apply Rle_ge; do 2 rewrite <- (Rplus_comm (- x0)); + apply Rplus_le_compat_l. +apply tech9; [ assumption | apply le_max_l ]. +left; assumption. +unfold ge in |- *; apply le_max_r. +apply Rplus_lt_reg_r with x0. +rewrite Rplus_0_r; unfold Rminus in |- *; rewrite (Rplus_comm x0); + rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r; + apply H4; apply le_n. +intros; apply Rlt_le_trans with (Un n). +case (Rlt_le_dec x0 (Un n)); intro. +assumption. +elim n0; assumption. +apply tech9; assumption. +unfold bound in |- *; exists x; unfold is_lub in H0; elim H0; intros; + assumption. +Qed. + +Lemma tech13 : + forall (An:nat -> R) (k:R), + 0 <= k < 1 -> + Un_cv (fun n:nat => Rabs (An (S n) / An n)) k -> + exists k0 : R, + k < k0 < 1 /\ + (exists N : nat, + (forall n:nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)). +intros; exists (k + (1 - k) / 2). +split. +split. +pattern k at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l. +unfold Rdiv in |- *; apply Rmult_lt_0_compat. +apply Rplus_lt_reg_r with k; rewrite Rplus_0_r; replace (k + (1 - k)) with 1; + [ elim H; intros; assumption | ring ]. +apply Rinv_0_lt_compat; prove_sup0. +apply Rmult_lt_reg_l with 2. +prove_sup0. +unfold Rdiv in |- *; rewrite Rmult_1_r; rewrite Rmult_plus_distr_l; + pattern 2 at 1 in |- *; rewrite Rmult_comm; rewrite Rmult_assoc; + rewrite <- Rinv_l_sym; [ idtac | discrR ]; rewrite Rmult_1_r; + replace (2 * k + (1 - k)) with (1 + k); [ idtac | ring ]. +elim H; intros. +apply Rplus_lt_compat_l; assumption. +unfold Un_cv in H0; cut (0 < (1 - k) / 2). +intro; elim (H0 ((1 - k) / 2) H1); intros. +exists x; intros. +assert (H4 := H2 n H3). +unfold R_dist in H4; rewrite <- Rabs_Rabsolu; + replace (Rabs (An (S n) / An n)) with (Rabs (An (S n) / An n) - k + k); + [ idtac | ring ]; + apply Rle_lt_trans with (Rabs (Rabs (An (S n) / An n) - k) + Rabs k). +apply Rabs_triang. +rewrite (Rabs_right k). +apply Rplus_lt_reg_r with (- k); rewrite <- (Rplus_comm k); + repeat rewrite <- Rplus_assoc; rewrite Rplus_opp_l; + repeat rewrite Rplus_0_l; apply H4. +apply Rle_ge; elim H; intros; assumption. +unfold Rdiv in |- *; apply Rmult_lt_0_compat. +apply Rplus_lt_reg_r with k; rewrite Rplus_0_r; elim H; intros; + replace (k + (1 - k)) with 1; [ assumption | ring ]. +apply Rinv_0_lt_compat; prove_sup0. +Qed. + +(**********) +Lemma growing_ineq : + forall (Un:nat -> R) (l:R), + Un_growing Un -> Un_cv Un l -> forall n:nat, Un n <= l. +intros; case (total_order_T (Un n) l); intro. +elim s; intro. +left; assumption. +right; assumption. +cut (0 < Un n - l). +intro; unfold Un_cv in H0; unfold R_dist in H0. +elim (H0 (Un n - l) H1); intros N1 H2. +set (N := max n N1). +cut (Un n - l <= Un N - l). +intro; cut (Un N - l < Un n - l). +intro; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H3 H4)). +apply Rle_lt_trans with (Rabs (Un N - l)). +apply RRle_abs. +apply H2. +unfold ge, N in |- *; apply le_max_r. +unfold Rminus in |- *; do 2 rewrite <- (Rplus_comm (- l)); + apply Rplus_le_compat_l. +apply tech9. +assumption. +unfold N in |- *; apply le_max_l. +apply Rplus_lt_reg_r with l. +rewrite Rplus_0_r. +replace (l + (Un n - l)) with (Un n); [ assumption | ring ]. +Qed. + +(* Un->l => (-Un) -> (-l) *) +Lemma CV_opp : + forall (An:nat -> R) (l:R), Un_cv An l -> Un_cv (opp_seq An) (- l). +intros An l. +unfold Un_cv in |- *; unfold R_dist in |- *; intros. +elim (H eps H0); intros. +exists x; intros. +unfold opp_seq in |- *; replace (- An n - - l) with (- (An n - l)); + [ rewrite Rabs_Ropp | ring ]. +apply H1; assumption. +Qed. + +(**********) +Lemma decreasing_ineq : + forall (Un:nat -> R) (l:R), + Un_decreasing Un -> Un_cv Un l -> forall n:nat, l <= Un n. +intros. +assert (H1 := decreasing_growing _ H). +assert (H2 := CV_opp _ _ H0). +assert (H3 := growing_ineq _ _ H1 H2). +apply Ropp_le_cancel. +unfold opp_seq in H3; apply H3. +Qed. + +(**********) +Lemma CV_minus : + forall (An Bn:nat -> R) (l1 l2:R), + Un_cv An l1 -> Un_cv Bn l2 -> Un_cv (fun i:nat => An i - Bn i) (l1 - l2). +intros. +replace (fun i:nat => An i - Bn i) with (fun i:nat => An i + opp_seq Bn i). +unfold Rminus in |- *; apply CV_plus. +assumption. +apply CV_opp; assumption. +unfold Rminus, opp_seq in |- *; reflexivity. +Qed. + +(* Un -> +oo *) +Definition cv_infty (Un:nat -> R) : Prop := + forall M:R, exists N : nat, (forall n:nat, (N <= n)%nat -> M < Un n). + +(* Un -> +oo => /Un -> O *) +Lemma cv_infty_cv_R0 : + forall Un:nat -> R, + (forall n:nat, Un n <> 0) -> cv_infty Un -> Un_cv (fun n:nat => / Un n) 0. +unfold cv_infty, Un_cv in |- *; unfold R_dist in |- *; intros. +elim (H0 (/ eps)); intros N0 H2. +exists N0; intros. +unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; + rewrite (Rabs_Rinv _ (H n)). +apply Rmult_lt_reg_l with (Rabs (Un n)). +apply Rabs_pos_lt; apply H. +rewrite <- Rinv_r_sym. +apply Rmult_lt_reg_l with (/ eps). +apply Rinv_0_lt_compat; assumption. +rewrite Rmult_1_r; rewrite (Rmult_comm (/ eps)); rewrite Rmult_assoc; + rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; apply Rlt_le_trans with (Un n). +apply H2; assumption. +apply RRle_abs. +red in |- *; intro; rewrite H4 in H1; elim (Rlt_irrefl _ H1). +apply Rabs_no_R0; apply H. +Qed. + +(**********) +Lemma decreasing_prop : + forall (Un:nat -> R) (m n:nat), + Un_decreasing Un -> (m <= n)%nat -> Un n <= Un m. +unfold Un_decreasing in |- *; intros. +induction n as [| n Hrecn]. +induction m as [| m Hrecm]. +right; reflexivity. +elim (le_Sn_O _ H0). +cut ((m <= n)%nat \/ m = S n). +intro; elim H1; intro. +apply Rle_trans with (Un n). +apply H. +apply Hrecn; assumption. +rewrite H2; right; reflexivity. +inversion H0; [ right; reflexivity | left; assumption ]. +Qed. + +(* |x|^n/n! -> 0 *) +Lemma cv_speed_pow_fact : + forall x:R, Un_cv (fun n:nat => x ^ n / INR (fact n)) 0. +intro; + cut + (Un_cv (fun n:nat => Rabs x ^ n / INR (fact n)) 0 -> + Un_cv (fun n:nat => x ^ n / INR (fact n)) 0). +intro; apply H. +unfold Un_cv in |- *; unfold R_dist in |- *; intros; case (Req_dec x 0); + intro. +exists 1%nat; intros. +rewrite H1; unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; + rewrite Rabs_R0; rewrite pow_ne_zero; + [ unfold Rdiv in |- *; rewrite Rmult_0_l; rewrite Rabs_R0; assumption + | red in |- *; intro; rewrite H3 in H2; elim (le_Sn_n _ H2) ]. +assert (H2 := Rabs_pos_lt x H1); set (M := up (Rabs x)); cut (0 <= M)%Z. +intro; elim (IZN M H3); intros M_nat H4. +set (Un := fun n:nat => Rabs x ^ (M_nat + n) / INR (fact (M_nat + n))). +cut (Un_cv Un 0); unfold Un_cv in |- *; unfold R_dist in |- *; intros. +elim (H5 eps H0); intros N H6. +exists (M_nat + N)%nat; intros; + cut (exists p : nat, (p >= N)%nat /\ n = (M_nat + p)%nat). +intro; elim H8; intros p H9. +elim H9; intros; rewrite H11; unfold Un in H6; apply H6; assumption. +exists (n - M_nat)%nat. +split. +unfold ge in |- *; apply (fun p n m:nat => plus_le_reg_l n m p) with M_nat; + rewrite <- le_plus_minus. +assumption. +apply le_trans with (M_nat + N)%nat. +apply le_plus_l. +assumption. +apply le_plus_minus; apply le_trans with (M_nat + N)%nat; + [ apply le_plus_l | assumption ]. +set (Vn := fun n:nat => Rabs x * (Un 0%nat / INR (S n))). +cut (1 <= M_nat)%nat. +intro; cut (forall n:nat, 0 < Un n). +intro; cut (Un_decreasing Un). +intro; cut (forall n:nat, Un (S n) <= Vn n). +intro; cut (Un_cv Vn 0). +unfold Un_cv in |- *; unfold R_dist in |- *; intros. +elim (H10 eps0 H5); intros N1 H11. +exists (S N1); intros. +cut (forall n:nat, 0 < Vn n). +intro; apply Rle_lt_trans with (Rabs (Vn (pred n) - 0)). +repeat rewrite Rabs_right. +unfold Rminus in |- *; rewrite Ropp_0; do 2 rewrite Rplus_0_r; + replace n with (S (pred n)). +apply H9. +inversion H12; simpl in |- *; reflexivity. +apply Rle_ge; unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; left; + apply H13. +apply Rle_ge; unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; left; + apply H7. +apply H11; unfold ge in |- *; apply le_S_n; replace (S (pred n)) with n; + [ unfold ge in H12; exact H12 | inversion H12; simpl in |- *; reflexivity ]. +intro; apply Rlt_le_trans with (Un (S n0)); [ apply H7 | apply H9 ]. +cut (cv_infty (fun n:nat => INR (S n))). +intro; cut (Un_cv (fun n:nat => / INR (S n)) 0). +unfold Un_cv, R_dist in |- *; intros; unfold Vn in |- *. +cut (0 < eps1 / (Rabs x * Un 0%nat)). +intro; elim (H11 _ H13); intros N H14. +exists N; intros; + replace (Rabs x * (Un 0%nat / INR (S n)) - 0) with + (Rabs x * Un 0%nat * (/ INR (S n) - 0)); + [ idtac | unfold Rdiv in |- *; ring ]. +rewrite Rabs_mult; apply Rmult_lt_reg_l with (/ Rabs (Rabs x * Un 0%nat)). +apply Rinv_0_lt_compat; apply Rabs_pos_lt. +apply prod_neq_R0. +apply Rabs_no_R0; assumption. +assert (H16 := H7 0%nat); red in |- *; intro; rewrite H17 in H16; + elim (Rlt_irrefl _ H16). +rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_l. +replace (/ Rabs (Rabs x * Un 0%nat) * eps1) with (eps1 / (Rabs x * Un 0%nat)). +apply H14; assumption. +unfold Rdiv in |- *; rewrite (Rabs_right (Rabs x * Un 0%nat)). +apply Rmult_comm. +apply Rle_ge; apply Rmult_le_pos. +apply Rabs_pos. +left; apply H7. +apply Rabs_no_R0. +apply prod_neq_R0; + [ apply Rabs_no_R0; assumption + | assert (H16 := H7 0%nat); red in |- *; intro; rewrite H17 in H16; + elim (Rlt_irrefl _ H16) ]. +unfold Rdiv in |- *; apply Rmult_lt_0_compat. +assumption. +apply Rinv_0_lt_compat; apply Rmult_lt_0_compat. +apply Rabs_pos_lt; assumption. +apply H7. +apply (cv_infty_cv_R0 (fun n:nat => INR (S n))). +intro; apply not_O_INR; discriminate. +assumption. +unfold cv_infty in |- *; intro; case (total_order_T M0 0); intro. +elim s; intro. +exists 0%nat; intros. +apply Rlt_trans with 0; [ assumption | apply lt_INR_0; apply lt_O_Sn ]. +exists 0%nat; intros; rewrite b; apply lt_INR_0; apply lt_O_Sn. +set (M0_z := up M0). +assert (H10 := archimed M0). +cut (0 <= M0_z)%Z. +intro; elim (IZN _ H11); intros M0_nat H12. +exists M0_nat; intros. +apply Rlt_le_trans with (IZR M0_z). +elim H10; intros; assumption. +rewrite H12; rewrite <- INR_IZR_INZ; apply le_INR. +apply le_trans with n; [ assumption | apply le_n_Sn ]. +apply le_IZR; left; simpl in |- *; unfold M0_z in |- *; + apply Rlt_trans with M0; [ assumption | elim H10; intros; assumption ]. +intro; apply Rle_trans with (Rabs x * Un n * / INR (S n)). +unfold Un in |- *; replace (M_nat + S n)%nat with (M_nat + n + 1)%nat. +rewrite pow_add; replace (Rabs x ^ 1) with (Rabs x); + [ idtac | simpl in |- *; ring ]. +unfold Rdiv in |- *; rewrite <- (Rmult_comm (Rabs x)); + repeat rewrite Rmult_assoc; repeat apply Rmult_le_compat_l. +apply Rabs_pos. +left; apply pow_lt; assumption. +replace (M_nat + n + 1)%nat with (S (M_nat + n)). +rewrite fact_simpl; rewrite mult_comm; rewrite mult_INR; + rewrite Rinv_mult_distr. +apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; apply lt_INR_0; apply neq_O_lt; red in |- *; + intro; assert (H10 := sym_eq H9); elim (fact_neq_0 _ H10). +left; apply Rinv_lt_contravar. +apply Rmult_lt_0_compat; apply lt_INR_0; apply lt_O_Sn. +apply lt_INR; apply lt_n_S. +pattern n at 1 in |- *; replace n with (0 + n)%nat; [ idtac | reflexivity ]. +apply plus_lt_compat_r. +apply lt_le_trans with 1%nat; [ apply lt_O_Sn | assumption ]. +apply INR_fact_neq_0. +apply not_O_INR; discriminate. +apply INR_eq; rewrite S_INR; do 3 rewrite plus_INR; reflexivity. +apply INR_eq; do 3 rewrite plus_INR; do 2 rewrite S_INR; ring. +unfold Vn in |- *; rewrite Rmult_assoc; unfold Rdiv in |- *; + rewrite (Rmult_comm (Un 0%nat)); rewrite (Rmult_comm (Un n)). +repeat apply Rmult_le_compat_l. +apply Rabs_pos. +left; apply Rinv_0_lt_compat; apply lt_INR_0; apply lt_O_Sn. +apply decreasing_prop; [ assumption | apply le_O_n ]. +unfold Un_decreasing in |- *; intro; unfold Un in |- *. +replace (M_nat + S n)%nat with (M_nat + n + 1)%nat. +rewrite pow_add; unfold Rdiv in |- *; rewrite Rmult_assoc; + apply Rmult_le_compat_l. +left; apply pow_lt; assumption. +replace (Rabs x ^ 1) with (Rabs x); [ idtac | simpl in |- *; ring ]. +replace (M_nat + n + 1)%nat with (S (M_nat + n)). +apply Rmult_le_reg_l with (INR (fact (S (M_nat + n)))). +apply lt_INR_0; apply neq_O_lt; red in |- *; intro; assert (H9 := sym_eq H8); + elim (fact_neq_0 _ H9). +rewrite (Rmult_comm (Rabs x)); rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym. +rewrite Rmult_1_l. +rewrite fact_simpl; rewrite mult_INR; rewrite Rmult_assoc; + rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; apply Rle_trans with (INR M_nat). +left; rewrite INR_IZR_INZ. +rewrite <- H4; assert (H8 := archimed (Rabs x)); elim H8; intros; assumption. +apply le_INR; apply le_trans with (S M_nat); + [ apply le_n_Sn | apply le_n_S; apply le_plus_l ]. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply INR_eq; rewrite S_INR; do 3 rewrite plus_INR; reflexivity. +apply INR_eq; do 3 rewrite plus_INR; do 2 rewrite S_INR; ring. +intro; unfold Un in |- *; unfold Rdiv in |- *; apply Rmult_lt_0_compat. +apply pow_lt; assumption. +apply Rinv_0_lt_compat; apply lt_INR_0; apply neq_O_lt; red in |- *; intro; + assert (H8 := sym_eq H7); elim (fact_neq_0 _ H8). +clear Un Vn; apply INR_le; simpl in |- *. +induction M_nat as [| M_nat HrecM_nat]. +assert (H6 := archimed (Rabs x)); fold M in H6; elim H6; intros. +rewrite H4 in H7; rewrite <- INR_IZR_INZ in H7. +simpl in H7; elim (Rlt_irrefl _ (Rlt_trans _ _ _ H2 H7)). +replace 1 with (INR 1); [ apply le_INR | reflexivity ]; apply le_n_S; + apply le_O_n. +apply le_IZR; simpl in |- *; left; apply Rlt_trans with (Rabs x). +assumption. +elim (archimed (Rabs x)); intros; assumption. +unfold Un_cv in |- *; unfold R_dist in |- *; intros; elim (H eps H0); intros. +exists x0; intros; + apply Rle_lt_trans with (Rabs (Rabs x ^ n / INR (fact n) - 0)). +unfold Rminus in |- *; rewrite Ropp_0; do 2 rewrite Rplus_0_r; + rewrite (Rabs_right (Rabs x ^ n / INR (fact n))). +unfold Rdiv in |- *; rewrite Rabs_mult; rewrite (Rabs_right (/ INR (fact n))). +rewrite RPow_abs; right; reflexivity. +apply Rle_ge; left; apply Rinv_0_lt_compat; apply lt_INR_0; apply neq_O_lt; + red in |- *; intro; assert (H4 := sym_eq H3); elim (fact_neq_0 _ H4). +apply Rle_ge; unfold Rdiv in |- *; apply Rmult_le_pos. +case (Req_dec x 0); intro. +rewrite H3; rewrite Rabs_R0. +induction n as [| n Hrecn]; + [ simpl in |- *; left; apply Rlt_0_1 + | simpl in |- *; rewrite Rmult_0_l; right; reflexivity ]. +left; apply pow_lt; apply Rabs_pos_lt; assumption. +left; apply Rinv_0_lt_compat; apply lt_INR_0; apply neq_O_lt; red in |- *; + intro; assert (H4 := sym_eq H3); elim (fact_neq_0 _ H4). +apply H1; assumption. +Qed. |