feat(theories/analysis): add theorems about convergent sequences, functions, and continuity
This commit is contained in:
Rob Lewis
Leonardo de Moura
parent
ffed988a34
commit
f402f322aa
2 changed files with 253 additions and 2 deletions
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@ -5,7 +5,7 @@ Author: Jeremy Avigad
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Metric spaces.
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-/
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import data.real.complete data.pnat
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import data.real.complete data.pnat data.list.sort data.fin
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open nat real eq.ops classical
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structure metric_space [class] (M : Type) : Type :=
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@ -161,6 +161,74 @@ proposition converges_to_seq_offset_succ_iff (X : ℕ → M) (y : M) :
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((λ n, X (succ n)) ⟶ y in ℕ) ↔ (X ⟶ y in ℕ) :=
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iff.intro converges_to_seq_of_converges_to_seq_offset_succ !converges_to_seq_offset_succ
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section
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open list
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definition r_trans : transitive (@le ℝ _) := λ a b c, !le.trans
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definition r_refl : reflexive (@le ℝ _) := λ a, !le.refl
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theorem dec_prf_eq (P : Prop) (H1 H2 : decidable P) : H1 = H2 :=
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begin
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induction H1,
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induction H2,
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reflexivity,
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apply absurd a a_1,
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induction H2,
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apply absurd a_1 a,
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reflexivity
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end
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-- there's a very ugly part of this proof.
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proposition bounded_of_converges_seq {X : ℕ → M} {x : M} (H : X ⟶ x in ℕ) :
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∃ K : ℝ, ∀ n : ℕ, dist (X n) x ≤ K :=
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begin
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cases H zero_lt_one with N HN,
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cases em (N = 0),
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existsi 1,
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intro n,
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apply le_of_lt,
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apply HN,
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rewrite a,
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apply zero_le,
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let l := map (λ n : ℕ, -dist (X n) x) (upto N),
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have Hnenil : l ≠ nil, from (map_ne_nil_of_ne_nil _ (upto_ne_nil_of_ne_zero a)),
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existsi max (-list.min (λ a b : ℝ, le a b) l Hnenil) 1,
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intro n,
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have Hsmn : ∀ m : ℕ, m < N → dist (X m) x ≤ max (-list.min (λ a b : ℝ, le a b) l Hnenil) 1, begin
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intro m Hm,
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apply le.trans,
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rotate 1,
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apply le_max_left,
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note Hall := min_lemma real.le_total r_trans r_refl Hnenil,
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have Hmem : -dist (X m) x ∈ (map (λ (n : ℕ), -dist (X n) x) (upto N)), from mem_map _ (mem_upto_of_lt Hm),
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note Hallm' := of_mem_of_all Hmem Hall,
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apply le_neg_of_le_neg,
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esimp, esimp at Hallm',
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have Heqs : (λ (a b : real), classical.prop_decidable (@le.{1} real real.real_has_le a b))
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=
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(@decidable_le.{1} real
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(@decidable_linear_ordered_comm_group.to_decidable_linear_order.{1} real
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(@decidable_linear_ordered_comm_ring.to_decidable_linear_ordered_comm_group.{1} real
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(@discrete_linear_ordered_field.to_decidable_linear_ordered_comm_ring.{1} real
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real.discrete_linear_ordered_field)))),
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begin
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apply funext, intro, apply funext, intro,
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apply dec_prf_eq
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end,
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rewrite -Heqs,
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exact Hallm'
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end,
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cases em (n < N) with Elt Ege,
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apply Hsmn,
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exact Elt,
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apply le_of_lt,
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apply lt_of_lt_of_le,
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apply HN,
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apply le_of_not_gt Ege,
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apply le_max_right
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end
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end
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/- cauchy sequences -/
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definition cauchy (X : ℕ → M) : Prop :=
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@ -282,6 +350,24 @@ theorem cnv_real_of_cnv_nat {X : ℕ → M} {c : M} (H : ∀ n : ℕ, dist (X n)
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end
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end
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theorem all_conv_seqs_of_converges_to_at {f : M → N} {c : M} {l : N} (Hconv : f ⟶ l at c) :
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∀ X : ℕ → M, ((∀ n : ℕ, ((X n) ≠ c) ∧ (X ⟶ c in ℕ)) → ((λ n : ℕ, f (X n)) ⟶ l in ℕ)) :=
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begin
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intros X HX,
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rewrite [↑converges_to_at at Hconv, ↑converges_to_seq],
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intros ε Hε,
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cases Hconv Hε with δ Hδ,
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cases Hδ with Hδ1 Hδ2,
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cases HX 0 with _ HXlim,
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cases HXlim Hδ1 with N1 HN1,
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existsi N1,
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intro n Hn,
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apply Hδ2,
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split,
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apply and.left (HX _),
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exact HN1 Hn
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end
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theorem converges_to_at_of_all_conv_seqs {f : M → N} (c : M) (l : N)
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(Hseq : ∀ X : ℕ → M, ((∀ n : ℕ, ((X n) ≠ c) ∧ (X ⟶ c in ℕ)) → ((λ n : ℕ, f (X n)) ⟶ l in ℕ)))
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: f ⟶ l at c :=
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@ -360,7 +446,7 @@ theorem converges_to_at_of_all_conv_seqs {f : M → N} (c : M) (l : N)
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definition continuous (f : M → N) : Prop := ∀ x, continuous_at f x
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theorem converges_seq_of_comp [instance] (X : ℕ → M) [HX : converges_seq X] {f : M → N}
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theorem converges_seq_comp_of_converges_seq_of_cts [instance] (X : ℕ → M) [HX : converges_seq X] {f : M → N}
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(Hf : continuous f) :
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converges_seq (λ n, f (X n)) :=
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begin
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@ -263,6 +263,141 @@ proposition mul_right_converges_to_seq (c : ℝ) (HX : X ⟶ x in ℕ) :
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have (λ n, X n * c) = (λ n, c * X n), from funext (take x, !mul.comm),
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by+ rewrite [this, mul.comm]; apply mul_left_converges_to_seq c HX
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protected lemma add_half_quarter {a : ℝ} (H : a > 0) : a / 2 + a / 4 < a :=
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begin
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replace (4 : ℝ) with (2 : ℝ) + 2,
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have Hne : (2 + 2 : ℝ) ≠ 0, from ne_of_gt four_pos,
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krewrite (div_add_div _ _ two_ne_zero Hne),
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have Hnum : (2 + 2 + 2) / (2 * (2 + 2)) = (3 : ℝ) / 4, by norm_num,
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rewrite [{2 * a}mul.comm, -left_distrib, mul_div_assoc, -mul_one a at {2}], krewrite Hnum,
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apply mul_lt_mul_of_pos_left,
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apply div_lt_of_mul_lt_of_pos,
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apply four_pos,
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rewrite one_mul,
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replace (3 : ℝ) with (2 : ℝ) + 1,
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replace (4 : ℝ) with (2 : ℝ) + 2,
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apply add_lt_add_left,
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apply two_gt_one,
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exact H
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end
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theorem converges_to_seq_squeeze (HX : X ⟶ x in ℕ) (HY : Y ⟶ x in ℕ) {Z : ℕ → ℝ} (HZX : ∀ n, X n ≤ Z n)
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(HZY : ∀ n, Z n ≤ Y n) : Z ⟶ x in ℕ :=
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begin
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intros ε Hε,
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have Hε4 : ε / 4 > 0, from div_pos_of_pos_of_pos Hε four_pos,
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cases HX Hε4 with N1 HN1,
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cases HY Hε4 with N2 HN2,
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existsi max N1 N2,
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intro n Hn,
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have HXY : abs (Y n - X n) < ε / 2, begin
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apply lt_of_le_of_lt,
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apply abs_sub_le _ x,
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have Hε24 : ε / 2 = ε / 4 + ε / 4, from eq.symm !add_quarters,
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rewrite Hε24,
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apply add_lt_add,
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apply HN2,
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apply ge.trans Hn !le_max_right,
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rewrite abs_sub,
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apply HN1,
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apply ge.trans Hn !le_max_left
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end,
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have HZX : abs (Z n - X n) < ε / 2, begin
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have HZXnp : Z n - X n ≥ 0, from sub_nonneg_of_le !HZX,
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have HXYnp : Y n - X n ≥ 0, from sub_nonneg_of_le (le.trans !HZX !HZY),
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rewrite [abs_of_nonneg HZXnp, abs_of_nonneg HXYnp at HXY],
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note Hgt := lt_add_of_sub_lt_right HXY,
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have Hlt : Z n < ε / 2 + X n, from calc
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Z n ≤ Y n : HZY
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... < ε / 2 + X n : Hgt,
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apply sub_lt_right_of_lt_add Hlt
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end,
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have H : abs (Z n - x) < ε, begin
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apply lt_of_le_of_lt,
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apply abs_sub_le _ (X n),
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apply lt.trans,
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apply add_lt_add,
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apply HZX,
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apply HN1,
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apply ge.trans Hn !le_max_left,
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apply add_half_quarter Hε
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end,
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exact H
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end
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proposition converges_to_seq_of_abs_sub_converges_to_seq (Habs : (λ n, abs (X n - x)) ⟶ 0 in ℕ) :
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X ⟶ x in ℕ :=
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begin
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intros ε Hε,
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cases Habs Hε with N HN,
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existsi N,
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intro n Hn,
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have Hn' : abs (abs (X n - x) - 0) < ε, from HN Hn,
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rewrite [sub_zero at Hn', abs_abs at Hn'],
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exact Hn'
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end
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proposition abs_sub_converges_to_seq_of_converges_to_seq (HX : X ⟶ x in ℕ) :
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(λ n, abs (X n - x)) ⟶ 0 in ℕ :=
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begin
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intros ε Hε,
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cases HX Hε with N HN,
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existsi N,
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intro n Hn,
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have Hn' : abs (abs (X n - x) - 0) < ε, by rewrite [sub_zero, abs_abs]; apply HN Hn,
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exact Hn'
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end
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proposition mul_converges_to_seq (HX : X ⟶ x in ℕ) (HY : Y ⟶ y in ℕ) :
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(λ n, X n * Y n) ⟶ x * y in ℕ :=
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begin
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have Hbd : ∃ K : ℝ, ∀ n : ℕ, abs (X n) ≤ K, begin
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cases bounded_of_converges_seq HX with K HK,
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existsi K + abs x,
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intro n,
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note Habs := le.trans (abs_abs_sub_abs_le_abs_sub (X n) x) !HK,
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apply le_add_of_sub_right_le,
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apply le.trans,
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apply le_abs_self,
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assumption
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end,
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cases Hbd with K HK,
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have Habsle : ∀ n, abs (X n * Y n - x * y) ≤ K * abs (Y n - y) + abs y * abs (X n - x), begin
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intro,
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have Heq : X n * Y n - x * y = (X n * Y n - X n * y) + (X n * y - x * y), by
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rewrite [-sub_add_cancel (X n * Y n) (X n * y) at {1}, sub_eq_add_neg, *add.assoc],
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apply le.trans,
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rewrite Heq,
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apply abs_add_le_abs_add_abs,
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apply add_le_add,
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rewrite [-mul_sub_left_distrib, abs_mul],
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apply mul_le_mul_of_nonneg_right,
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apply HK,
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apply abs_nonneg,
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rewrite [-mul_sub_right_distrib, abs_mul, mul.comm],
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apply le.refl
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end,
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have Hdifflim : (λ n, abs (X n * Y n - x * y)) ⟶ 0 in ℕ, begin
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apply converges_to_seq_squeeze,
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rotate 2,
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intro, apply abs_nonneg,
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apply Habsle,
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apply converges_to_seq_constant,
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rewrite -{0}zero_add,
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apply add_converges_to_seq,
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rewrite -(mul_zero K),
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apply mul_left_converges_to_seq,
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apply abs_sub_converges_to_seq_of_converges_to_seq,
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exact HY,
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rewrite -(mul_zero (abs y)),
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apply mul_left_converges_to_seq,
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apply abs_sub_converges_to_seq_of_converges_to_seq,
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exact HX
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end,
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apply converges_to_seq_of_abs_sub_converges_to_seq,
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apply Hdifflim
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end
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-- TODO: converges_to_seq_div, converges_to_seq_mul_left_iff, etc.
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proposition abs_converges_to_seq_zero (HX : X ⟶ 0 in ℕ) : (λ n, abs (X n)) ⟶ 0 in ℕ :=
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@ -280,6 +415,25 @@ iff.intro converges_to_seq_zero_of_abs_converges_to_seq_zero abs_converges_to_se
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end limit_operations
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/- properties of converges_to_at -/
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section limit_operations_continuous
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variables {f g : ℝ → ℝ}
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variables {a b x y : ℝ}
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theorem mul_converges_to_at (Hf : f ⟶ a at x) (Hg : g ⟶ b at x) : (λ z, f z * g z) ⟶ a * b at x :=
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begin
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apply converges_to_at_of_all_conv_seqs,
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intro X HX,
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apply mul_converges_to_seq,
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note Hfc := all_conv_seqs_of_converges_to_at Hf,
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apply Hfc _ HX,
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note Hgb := all_conv_seqs_of_converges_to_at Hg,
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apply Hgb _ HX
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end
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end limit_operations_continuous
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/- monotone sequences -/
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section monotone_sequences
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@ -487,6 +641,17 @@ theorem continuous_offset_of_continuous {f : ℝ → ℝ} (Hcon : continuous f)
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assumption
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end
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theorem continuous_mul_of_continuous {f g : ℝ → ℝ} (Hconf : continuous f) (Hcong : continuous g) :
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continuous (λ x, f x * g x) :=
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begin
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intro x,
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apply continuous_at_of_converges_to_at,
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apply mul_converges_to_at,
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all_goals apply converges_to_at_of_continuous_at,
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apply Hconf,
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apply Hcong
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end
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end continuous
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/-
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