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f1f7953 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 | 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
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