| import Mathlib.Analysis.InnerProductSpace.Basic |
| import Mathlib.Topology.Instances.Real.Lemmas |
| import Mathlib.Topology.MetricSpace.Cauchy |
|
|
| namespace OAI |
|
|
| open Filter |
| open scoped Topology |
|
|
| namespace Invariant233.ProductDensity |
|
|
| theorem cauchySeq_of_prefix_inner |
| {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| (f : ℕ → E) |
| (hinner : ∀ n m, n ≤ m → inner ℝ (f n) (f m) = ‖f n‖ ^ 2) |
| (hbound : BddAbove (Set.range fun n => ‖f n‖ ^ 2)) : CauchySeq f := by |
| have hdist : ∀ n m, n ≤ m → dist (f n) (f m) ^ 2 = ‖f m‖ ^ 2 - ‖f n‖ ^ 2 := by |
| intro n m hnm |
| rw [dist_eq_norm, norm_sub_sq_real, hinner n m hnm] |
| ring |
| have hmono : Monotone (fun n => ‖f n‖ ^ 2) := by |
| intro n m hnm |
| have := sq_nonneg (dist (f n) (f m)) |
| rw [hdist n m hnm] at this |
| linarith |
| have hc : CauchySeq (fun n => ‖f n‖ ^ 2) := |
| (tendsto_atTop_ciSup hmono hbound).cauchySeq |
| apply Metric.cauchySeq_iff.2 |
| intro ε hε |
| obtain ⟨N, hN⟩ := Metric.cauchySeq_iff.1 hc (ε ^ 2) (sq_pos_of_pos hε) |
| refine ⟨N, fun n hn m hm => ?_⟩ |
| have hreal := hN n hn m hm |
| rw [Real.dist_eq] at hreal |
| rcases le_total n m with hnm | hmn |
| · have hs := hdist n m hnm |
| have hl := (abs_lt.mp hreal).1 |
| nlinarith [dist_nonneg (x := f n) (y := f m)] |
| · have hs := hdist m n hmn |
| have hl := (abs_lt.mp hreal).2 |
| rw [dist_comm (f m) (f n)] at hs |
| nlinarith [dist_nonneg (x := f n) (y := f m)] |
|
|
| theorem exists_limit_of_prefix_inner |
| {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] |
| (f : ℕ → E) |
| (hinner : ∀ n m, n ≤ m → inner ℝ (f n) (f m) = ‖f n‖ ^ 2) |
| (hbound : BddAbove (Set.range fun n => ‖f n‖ ^ 2)) : |
| ∃ g : E, Tendsto f atTop (𝓝 g) := |
| cauchySeq_tendsto_of_complete (cauchySeq_of_prefix_inner f hinner hbound) |
|
|
| end Invariant233.ProductDensity |
|
|
| end OAI |
|
|
