CE_strictMono

formal source

  71  (bounded_subseq_tendsto (fun n => f (CE f c R hf m n) (m + 1)) c R
  72    (fun n => hf (CE f c R hf m n) (m + 1))).choose_spec.choose_spec.1
  73
  74private theorem CE_strictMono : ∀ m, StrictMono (CE f c R hf m) := by
  75  intro m; induction m with
  76  | zero =>
  77    exact (bounded_subseq_tendsto (fun n => f n 0) c R (fun n => hf n 0)).choose_spec.choose_spec.1
  78  | succ k ih =>
  79    rw [CE_succ]; exact ih.comp (LE_strictMono f c R hf k)
  80
  81private theorem LE_conv (m : ℕ) :
  82    Tendsto (fun n => f (CE f c R hf m (LE f c R hf m n)) (m + 1)) atTop
  83      (𝓝 (CL f c R hf (m + 1))) :=
  84  (bounded_subseq_tendsto (fun n => f (CE f c R hf m n) (m + 1)) c R
  85    (fun n => hf (CE f c R hf m n) (m + 1))).choose_spec.choose_spec.2
  86
  87private theorem CE_conv_at : ∀ m, Tendsto (fun n => f (CE f c R hf m n) m)
  88    atTop (𝓝 (CL f c R hf m)) := by
  89  intro m; cases m with
  90  | zero =>
  91    exact (bounded_subseq_tendsto (fun n => f n 0) c R (fun n => hf n 0)).choose_spec.choose_spec.2
  92  | succ k => exact LE_conv f c R hf k
  93
  94private theorem CE_conv_le : ∀ m j, j ≤ m →
  95    Tendsto (fun n => f (CE f c R hf m n) j) atTop (𝓝 (CL f c R hf j)) := by
  96  intro m j hj
  97  induction m with
  98  | zero =>
  99    have hj0 : j = 0 := Nat.le_zero.mp hj
 100    subst hj0; exact CE_conv_at f c R hf 0
 101  | succ k ih =>
 102    rcases eq_or_lt_of_le hj with hjk | hjk
 103    · subst hjk; exact CE_conv_at f c R hf (k + 1)
 104    · rw [CE_succ]