| import Mathlib | |
| namespace OAI | |
| open Filter | |
| open scoped Topology | |
| namespace Invariant233.ProductDensity | |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] | |
| theorem norm_sub_sq_of_nested_inner (u : ℕ → E) | |
| (hinner : ∀ m n, m ≤ n → inner ℝ (u m) (u n) = ‖u m‖ ^ 2) | |
| {m n : ℕ} (hmn : m ≤ n) : | |
| ‖u m - u n‖ ^ 2 = ‖u n‖ ^ 2 - ‖u m‖ ^ 2 := by | |
| rw [norm_sub_sq_real, hinner m n hmn] | |
| ring | |
| theorem monotone_norm_sq_of_nested_inner (u : ℕ → E) | |
| (hinner : ∀ m n, m ≤ n → inner ℝ (u m) (u n) = ‖u m‖ ^ 2) : | |
| Monotone (fun n => ‖u n‖ ^ 2) := by | |
| intro m n hmn | |
| have h := norm_sub_sq_of_nested_inner u hinner hmn | |
| nlinarith [sq_nonneg ‖u m - u n‖] | |
| theorem norm_sub_sq_eq_abs_of_nested_inner (u : ℕ → E) | |
| (hinner : ∀ m n, m ≤ n → inner ℝ (u m) (u n) = ‖u m‖ ^ 2) | |
| (m n : ℕ) : | |
| ‖u m - u n‖ ^ 2 = |‖u m‖ ^ 2 - ‖u n‖ ^ 2| := by | |
| rcases le_total m n with hmn | hnm | |
| · rw [norm_sub_sq_of_nested_inner u hinner hmn, | |
| abs_of_nonpos (sub_nonpos.mpr (monotone_norm_sq_of_nested_inner u hinner hmn))] | |
| ring | |
| · rw [norm_sub_rev, norm_sub_sq_of_nested_inner u hinner hnm, | |
| abs_of_nonneg (sub_nonneg.mpr (monotone_norm_sq_of_nested_inner u hinner hnm))] | |
| theorem cauchySeq_of_nested_inner (u : ℕ → E) | |
| (hinner : ∀ m n, m ≤ n → inner ℝ (u m) (u n) = ‖u m‖ ^ 2) | |
| (hbound : BddAbove (Set.range (fun n => ‖u n‖ ^ 2))) : | |
| CauchySeq u := by | |
| have hlim := tendsto_atTop_ciSup (monotone_norm_sq_of_nested_inner u hinner) hbound | |
| have hc : CauchySeq (fun n => ‖u n‖ ^ 2) := hlim.cauchySeq | |
| apply Metric.cauchySeq_iff.mpr | |
| intro ε hε | |
| obtain ⟨N, hN⟩ := Metric.cauchySeq_iff.mp hc (ε ^ 2) (sq_pos_of_pos hε) | |
| refine ⟨N, ?_⟩ | |
| intro m hm n hn | |
| have hs := hN m hm n hn | |
| rw [Real.dist_eq, ← norm_sub_sq_eq_abs_of_nested_inner u hinner m n] at hs | |
| rw [dist_eq_norm] | |
| nlinarith [norm_nonneg (u m - u n)] | |
| theorem exists_limit_of_nested_inner [CompleteSpace E] (u : ℕ → E) | |
| (hinner : ∀ m n, m ≤ n → inner ℝ (u m) (u n) = ‖u m‖ ^ 2) | |
| (C : ℝ) (hbound : ∀ n, ‖u n‖ ^ 2 ≤ C) : | |
| ∃ v : E, Tendsto u atTop (𝓝 v) ∧ ‖v‖ ^ 2 ≤ C := by | |
| obtain ⟨v, hv⟩ := cauchySeq_tendsto_of_complete | |
| (cauchySeq_of_nested_inner u hinner ⟨C, by rintro _ ⟨n, rfl⟩; exact hbound n⟩) | |
| refine ⟨v, hv, ?_⟩ | |
| exact le_of_tendsto (hv.norm.pow 2) (Eventually.of_forall hbound) | |
| end Invariant233.ProductDensity | |
| end OAI | |