11.5. HardyZeta.FiniteDifferenceFlat🔗

theorem tendsto_scaled_boundary_value {f : ℝ → ℝ} {a : ℝ} (n : ℕ)
    (hf : Tendsto (fun u : ℝ ↦ u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0))
    {c : ℝ} (hc : 0 < c) :
    Tendsto (fun h : ℝ ↦ h⁻¹ ^ n * f (a - c * h)) (𝓝[>] 0) (𝓝 0) := byf:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)c:ℝhc:0 < c⊢ Tendsto (fun h => h⁻¹ ^ n * f (a - c * h)) (𝓝[>] 0) (𝓝 0)
  have hm : Tendsto (fun h : ℝ ↦ c * h) (𝓝[>] 0) (𝓝[>] 0) := by
    apply tendsto_nhdsWithin_iff.mprf:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)c:ℝhc:0 < c⊢ Tendsto (fun h => c * h) (𝓝[>] 0) (𝓝 0) ∧ ∀ᶠ (n : ℝ) in 𝓝[>] 0, c * n ∈ Ioi 0
    constructorleftf:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)c:ℝhc:0 < c⊢ Tendsto (fun h => c * h) (𝓝[>] 0) (𝓝 0)rightf:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)c:ℝhc:0 < c⊢ ∀ᶠ (n : ℝ) in 𝓝[>] 0, c * n ∈ Ioi 0
    ·leftf:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)c:ℝhc:0 < c⊢ Tendsto (fun h => c * h) (𝓝[>] 0) (𝓝 0) have hid : Tendsto (fun h : ℝ ↦ h) (𝓝[>] 0) (𝓝 0) := nhdsWithin_le_nhdsleftf:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)c:ℝhc:0 < chid:Tendsto (fun h => h) (𝓝[>] 0) (𝓝 0)⊢ Tendsto (fun h => c * h) (𝓝[>] 0) (𝓝 0)
      simpa using hid.const_mul cAll goals completed! 🐙
    ·rightf:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)c:ℝhc:0 < c⊢ ∀ᶠ (n : ℝ) in 𝓝[>] 0, c * n ∈ Ioi 0 filter_upwards [self_mem_nhdsWithin] with h hhf:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)c:ℝhc:0 < ch:ℝhh:h ∈ Ioi 0⊢ c * h ∈ Ioi 0 using mul_pos hc hhf:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)c:ℝhc:0 < chm:Tendsto (fun h => c * h) (𝓝[>] 0) (𝓝[>] 0)⊢ Tendsto (fun h => h⁻¹ ^ n * f (a - c * h)) (𝓝[>] 0) (𝓝 0)
  have hl := (hf.comp hm).const_mul (c ^ n)f:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)c:ℝhc:0 < chm:Tendsto (fun h => c * h) (𝓝[>] 0) (𝓝[>] 0)hl:Tendsto (fun x => c ^ n * ((fun u => u⁻¹ ^ n * f (a - u)) ∘ fun h => c * h) x) (𝓝[>] 0) (𝓝 (c ^ n * 0))⊢ Tendsto (fun h => h⁻¹ ^ n * f (a - c * h)) (𝓝[>] 0) (𝓝 0)
  simp only [mul_zero] at hlf:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)c:ℝhc:0 < chm:Tendsto (fun h => c * h) (𝓝[>] 0) (𝓝[>] 0)hl:Tendsto (fun x => c ^ n * ((fun u => u⁻¹ ^ n * f (a - u)) ∘ fun h => c * h) x) (𝓝[>] 0) (𝓝 0)⊢ Tendsto (fun h => h⁻¹ ^ n * f (a - c * h)) (𝓝[>] 0) (𝓝 0)
  convert hl using 1f:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)c:ℝhc:0 < chm:Tendsto (fun h => c * h) (𝓝[>] 0) (𝓝[>] 0)hl:Tendsto (fun x => c ^ n * ((fun u => u⁻¹ ^ n * f (a - u)) ∘ fun h => c * h) x) (𝓝[>] 0) (𝓝 0)⊢ (fun h => h⁻¹ ^ n * f (a - c * h)) = fun x => c ^ n * ((fun u => u⁻¹ ^ n * f (a - u)) ∘ fun h => c * h) x
  funext hf:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)c:ℝhc:0 < chm:Tendsto (fun h => c * h) (𝓝[>] 0) (𝓝[>] 0)hl:Tendsto (fun x => c ^ n * ((fun u => u⁻¹ ^ n * f (a - u)) ∘ fun h => c * h) x) (𝓝[>] 0) (𝓝 0)h:ℝ⊢ h⁻¹ ^ n * f (a - c * h) = c ^ n * ((fun u => u⁻¹ ^ n * f (a - u)) ∘ fun h => c * h) h
  dsimpf:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)c:ℝhc:0 < chm:Tendsto (fun h => c * h) (𝓝[>] 0) (𝓝[>] 0)hl:Tendsto (fun x => c ^ n * ((fun u => u⁻¹ ^ n * f (a - u)) ∘ fun h => c * h) x) (𝓝[>] 0) (𝓝 0)h:ℝ⊢ h⁻¹ ^ n * f (a - c * h) = c ^ n * ((c * h)⁻¹ ^ n * f (a - c * h))
  have he : c ^ n * (c * h)⁻¹ ^ n = h⁻¹ ^ n := byf:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)c:ℝhc:0 < c⊢ Tendsto (fun h => h⁻¹ ^ n * f (a - c * h)) (𝓝[>] 0) (𝓝 0)
    rw [← mul_pow,f:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)c:ℝhc:0 < chm:Tendsto (fun h => c * h) (𝓝[>] 0) (𝓝[>] 0)hl:Tendsto (fun x => c ^ n * ((fun u => u⁻¹ ^ n * f (a - u)) ∘ fun h => c * h) x) (𝓝[>] 0) (𝓝 0)h:ℝ⊢ (c * (c * h)⁻¹) ^ n = h⁻¹ ^ n mul_inv_revf:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)c:ℝhc:0 < chm:Tendsto (fun h => c * h) (𝓝[>] 0) (𝓝[>] 0)hl:Tendsto (fun x => c ^ n * ((fun u => u⁻¹ ^ n * f (a - u)) ∘ fun h => c * h) x) (𝓝[>] 0) (𝓝 0)h:ℝ⊢ (c * (h⁻¹ * c⁻¹)) ^ n = h⁻¹ ^ n]f:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)c:ℝhc:0 < chm:Tendsto (fun h => c * h) (𝓝[>] 0) (𝓝[>] 0)hl:Tendsto (fun x => c ^ n * ((fun u => u⁻¹ ^ n * f (a - u)) ∘ fun h => c * h) x) (𝓝[>] 0) (𝓝 0)h:ℝ⊢ (c * (h⁻¹ * c⁻¹)) ^ n = h⁻¹ ^ n
    have he : c * (h⁻¹ * c⁻¹) = h⁻¹ := byf:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)c:ℝhc:0 < c⊢ Tendsto (fun h => h⁻¹ ^ n * f (a - c * h)) (𝓝[>] 0) (𝓝 0) field_simpf:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)c:ℝhc:0 < chm:Tendsto (fun h => c * h) (𝓝[>] 0) (𝓝[>] 0)hl:Tendsto (fun x => c ^ n * ((fun u => u⁻¹ ^ n * f (a - u)) ∘ fun h => c * h) x) (𝓝[>] 0) (𝓝 0)h:ℝhe:c * (h⁻¹ * c⁻¹) = h⁻¹⊢ (c * (h⁻¹ * c⁻¹)) ^ n = h⁻¹ ^ n
    rw [hef:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)c:ℝhc:0 < chm:Tendsto (fun h => c * h) (𝓝[>] 0) (𝓝[>] 0)hl:Tendsto (fun x => c ^ n * ((fun u => u⁻¹ ^ n * f (a - u)) ∘ fun h => c * h) x) (𝓝[>] 0) (𝓝 0)h:ℝhe:c * (h⁻¹ * c⁻¹) = h⁻¹⊢ h⁻¹ ^ n = h⁻¹ ^ n]f:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)c:ℝhc:0 < chm:Tendsto (fun h => c * h) (𝓝[>] 0) (𝓝[>] 0)hl:Tendsto (fun x => c ^ n * ((fun u => u⁻¹ ^ n * f (a - u)) ∘ fun h => c * h) x) (𝓝[>] 0) (𝓝 0)h:ℝhe:c ^ n * (c * h)⁻¹ ^ n = h⁻¹ ^ n⊢ h⁻¹ ^ n * f (a - c * h) = c ^ n * ((c * h)⁻¹ ^ n * f (a - c * h))
  rw [← mul_assoc,f:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)c:ℝhc:0 < chm:Tendsto (fun h => c * h) (𝓝[>] 0) (𝓝[>] 0)hl:Tendsto (fun x => c ^ n * ((fun u => u⁻¹ ^ n * f (a - u)) ∘ fun h => c * h) x) (𝓝[>] 0) (𝓝 0)h:ℝhe:c ^ n * (c * h)⁻¹ ^ n = h⁻¹ ^ n⊢ h⁻¹ ^ n * f (a - c * h) = c ^ n * (c * h)⁻¹ ^ n * f (a - c * h) hef:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)c:ℝhc:0 < chm:Tendsto (fun h => c * h) (𝓝[>] 0) (𝓝[>] 0)hl:Tendsto (fun x => c ^ n * ((fun u => u⁻¹ ^ n * f (a - u)) ∘ fun h => c * h) x) (𝓝[>] 0) (𝓝 0)h:ℝhe:c ^ n * (c * h)⁻¹ ^ n = h⁻¹ ^ n⊢ h⁻¹ ^ n * f (a - c * h) = h⁻¹ ^ n * f (a - c * h)]All goals completed! 🐙

/-- All samples stay inside the boundary when the center is more than k steps away. -/

theorem tendsto_scaled_iterate_modifiedDifference {f : ℝ → ℝ} {a : ℝ} (n k : ℕ)
    (hf : Tendsto (fun u : ℝ ↦ u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0))
    {c : ℝ} (hc : (k : ℝ) < c) :
    Tendsto (fun h : ℝ ↦ h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h))
      (𝓝[>] 0) (𝓝 0) := byf:ℝ → ℝa:ℝn:ℕk:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)c:ℝhc:↑k < c⊢ Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0)
  induction k generalizing c with
  | zero =>zerof:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)c:ℝhc:↑0 < c⊢ Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[0] f (a - c * h)) (𝓝[>] 0) (𝓝 0)
    simpa using tendsto_scaled_boundary_value n hf (byf:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)c:ℝhc:↑0 < c⊢ 0 < c simpa using hcAll goals completed! 🐙)
  | succ k ih =>succf:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)k:ℕih:∀ {c : ℝ}, ↑k < c → Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0)c:ℝhc:↑(k + 1) < c⊢ Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k + 1] f (a - c * h)) (𝓝[>] 0) (𝓝 0)
    push_cast at hcsuccf:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)k:ℕih:∀ {c : ℝ}, ↑k < c → Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0)c:ℝhc:↑k + 1 < c⊢ Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k + 1] f (a - c * h)) (𝓝[>] 0) (𝓝 0)
    have hp := ih (c := c - 1) (byf:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)k:ℕih:∀ {c : ℝ}, ↑k < c → Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0)c:ℝhc:↑k + 1 < c⊢ ↑k < c - 1 linarithAll goals completed! 🐙)succf:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)k:ℕih:∀ {c : ℝ}, ↑k < c → Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0)c:ℝhc:↑k + 1 < chp:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c - 1) * h)) (𝓝[>] 0) (𝓝 0)⊢ Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k + 1] f (a - c * h)) (𝓝[>] 0) (𝓝 0)
    have hm := ih (c := c + 1) (byf:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)k:ℕih:∀ {c : ℝ}, ↑k < c → Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0)c:ℝhc:↑k + 1 < chp:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c - 1) * h)) (𝓝[>] 0) (𝓝 0)⊢ ↑k < c + 1 linarithAll goals completed! 🐙)succf:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)k:ℕih:∀ {c : ℝ}, ↑k < c → Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0)c:ℝhc:↑k + 1 < chp:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c - 1) * h)) (𝓝[>] 0) (𝓝 0)hm:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c + 1) * h)) (𝓝[>] 0) (𝓝 0)⊢ Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k + 1] f (a - c * h)) (𝓝[>] 0) (𝓝 0)
    have h0 := ih (c := c) (byf:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)k:ℕih:∀ {c : ℝ}, ↑k < c → Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0)c:ℝhc:↑k + 1 < chp:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c - 1) * h)) (𝓝[>] 0) (𝓝 0)hm:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c + 1) * h)) (𝓝[>] 0) (𝓝 0)⊢ ↑k < c linarithAll goals completed! 🐙)succf:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)k:ℕih:∀ {c : ℝ}, ↑k < c → Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0)c:ℝhc:↑k + 1 < chp:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c - 1) * h)) (𝓝[>] 0) (𝓝 0)hm:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c + 1) * h)) (𝓝[>] 0) (𝓝 0)h0:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0)⊢ Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k + 1] f (a - c * h)) (𝓝[>] 0) (𝓝 0)
    have hcos : Tendsto (fun h : ℝ ↦ 2 * Real.cos (h / 2)) (𝓝[>] 0) (𝓝 2) := byf:ℝ → ℝa:ℝn:ℕk:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)c:ℝhc:↑k < c⊢ Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0)
      have h : ContinuousAt (fun h : ℝ ↦ 2 * Real.cos (h / 2)) 0 := byf:ℝ → ℝa:ℝn:ℕk:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)c:ℝhc:↑k < c⊢ Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0) fun_propf:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)k:ℕih:∀ {c : ℝ}, ↑k < c → Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0)c:ℝhc:↑k + 1 < chp:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c - 1) * h)) (𝓝[>] 0) (𝓝 0)hm:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c + 1) * h)) (𝓝[>] 0) (𝓝 0)h0:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0)h:ContinuousAt (fun h => 2 * Real.cos (h / 2)) 0⊢ Tendsto (fun h => 2 * Real.cos (h / 2)) (𝓝[>] 0) (𝓝 2)
      simpa using h.tendsto.mono_left nhdsWithin_le_nhdssuccf:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)k:ℕih:∀ {c : ℝ}, ↑k < c → Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0)c:ℝhc:↑k + 1 < chp:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c - 1) * h)) (𝓝[>] 0) (𝓝 0)hm:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c + 1) * h)) (𝓝[>] 0) (𝓝 0)h0:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0)hcos:Tendsto (fun h => 2 * Real.cos (h / 2)) (𝓝[>] 0) (𝓝 2)⊢ Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k + 1] f (a - c * h)) (𝓝[>] 0) (𝓝 0)
    have hl := (hp.add hm).sub (hcos.mul h0)succf:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)k:ℕih:∀ {c : ℝ}, ↑k < c → Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0)c:ℝhc:↑k + 1 < chp:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c - 1) * h)) (𝓝[>] 0) (𝓝 0)hm:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c + 1) * h)) (𝓝[>] 0) (𝓝 0)h0:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0)hcos:Tendsto (fun h => 2 * Real.cos (h / 2)) (𝓝[>] 0) (𝓝 2)hl:Tendsto
  (fun x =>
    x⁻¹ ^ n * (modifiedDifference x)^[k] f (a - (c - 1) * x) +
        x⁻¹ ^ n * (modifiedDifference x)^[k] f (a - (c + 1) * x) -
      2 * Real.cos (x / 2) * (x⁻¹ ^ n * (modifiedDifference x)^[k] f (a - c * x)))
  (𝓝[>] 0) (𝓝 (0 + 0 - 2 * 0))⊢ Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k + 1] f (a - c * h)) (𝓝[>] 0) (𝓝 0)
    simp only [add_zero, mul_zero, sub_zero] at hlsuccf:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)k:ℕih:∀ {c : ℝ}, ↑k < c → Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0)c:ℝhc:↑k + 1 < chp:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c - 1) * h)) (𝓝[>] 0) (𝓝 0)hm:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c + 1) * h)) (𝓝[>] 0) (𝓝 0)h0:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0)hcos:Tendsto (fun h => 2 * Real.cos (h / 2)) (𝓝[>] 0) (𝓝 2)hl:Tendsto
  (fun x =>
    x⁻¹ ^ n * (modifiedDifference x)^[k] f (a - (c - 1) * x) +
        x⁻¹ ^ n * (modifiedDifference x)^[k] f (a - (c + 1) * x) -
      2 * Real.cos (x / 2) * (x⁻¹ ^ n * (modifiedDifference x)^[k] f (a - c * x)))
  (𝓝[>] 0) (𝓝 0)⊢ Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k + 1] f (a - c * h)) (𝓝[>] 0) (𝓝 0)
    convert hl using 1f:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)k:ℕih:∀ {c : ℝ}, ↑k < c → Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0)c:ℝhc:↑k + 1 < chp:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c - 1) * h)) (𝓝[>] 0) (𝓝 0)hm:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c + 1) * h)) (𝓝[>] 0) (𝓝 0)h0:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0)hcos:Tendsto (fun h => 2 * Real.cos (h / 2)) (𝓝[>] 0) (𝓝 2)hl:Tendsto
  (fun x =>
    x⁻¹ ^ n * (modifiedDifference x)^[k] f (a - (c - 1) * x) +
        x⁻¹ ^ n * (modifiedDifference x)^[k] f (a - (c + 1) * x) -
      2 * Real.cos (x / 2) * (x⁻¹ ^ n * (modifiedDifference x)^[k] f (a - c * x)))
  (𝓝[>] 0) (𝓝 0)⊢ (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k + 1] f (a - c * h)) = fun x =>
  x⁻¹ ^ n * (modifiedDifference x)^[k] f (a - (c - 1) * x) + x⁻¹ ^ n * (modifiedDifference x)^[k] f (a - (c + 1) * x) -
    2 * Real.cos (x / 2) * (x⁻¹ ^ n * (modifiedDifference x)^[k] f (a - c * x))
    funext hf:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)k:ℕih:∀ {c : ℝ}, ↑k < c → Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0)c:ℝhc:↑k + 1 < chp:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c - 1) * h)) (𝓝[>] 0) (𝓝 0)hm:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c + 1) * h)) (𝓝[>] 0) (𝓝 0)h0:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0)hcos:Tendsto (fun h => 2 * Real.cos (h / 2)) (𝓝[>] 0) (𝓝 2)hl:Tendsto
  (fun x =>
    x⁻¹ ^ n * (modifiedDifference x)^[k] f (a - (c - 1) * x) +
        x⁻¹ ^ n * (modifiedDifference x)^[k] f (a - (c + 1) * x) -
      2 * Real.cos (x / 2) * (x⁻¹ ^ n * (modifiedDifference x)^[k] f (a - c * x)))
  (𝓝[>] 0) (𝓝 0)h:ℝ⊢ h⁻¹ ^ n * (modifiedDifference h)^[k + 1] f (a - c * h) =
  h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c - 1) * h) + h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c + 1) * h) -
    2 * Real.cos (h / 2) * (h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h))
    rw [Function.iterate_succ_apply'f:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)k:ℕih:∀ {c : ℝ}, ↑k < c → Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0)c:ℝhc:↑k + 1 < chp:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c - 1) * h)) (𝓝[>] 0) (𝓝 0)hm:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c + 1) * h)) (𝓝[>] 0) (𝓝 0)h0:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0)hcos:Tendsto (fun h => 2 * Real.cos (h / 2)) (𝓝[>] 0) (𝓝 2)hl:Tendsto
  (fun x =>
    x⁻¹ ^ n * (modifiedDifference x)^[k] f (a - (c - 1) * x) +
        x⁻¹ ^ n * (modifiedDifference x)^[k] f (a - (c + 1) * x) -
      2 * Real.cos (x / 2) * (x⁻¹ ^ n * (modifiedDifference x)^[k] f (a - c * x)))
  (𝓝[>] 0) (𝓝 0)h:ℝ⊢ h⁻¹ ^ n * modifiedDifference h ((modifiedDifference h)^[k] f) (a - c * h) =
  h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c - 1) * h) + h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c + 1) * h) -
    2 * Real.cos (h / 2) * (h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h))]f:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)k:ℕih:∀ {c : ℝ}, ↑k < c → Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0)c:ℝhc:↑k + 1 < chp:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c - 1) * h)) (𝓝[>] 0) (𝓝 0)hm:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c + 1) * h)) (𝓝[>] 0) (𝓝 0)h0:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0)hcos:Tendsto (fun h => 2 * Real.cos (h / 2)) (𝓝[>] 0) (𝓝 2)hl:Tendsto
  (fun x =>
    x⁻¹ ^ n * (modifiedDifference x)^[k] f (a - (c - 1) * x) +
        x⁻¹ ^ n * (modifiedDifference x)^[k] f (a - (c + 1) * x) -
      2 * Real.cos (x / 2) * (x⁻¹ ^ n * (modifiedDifference x)^[k] f (a - c * x)))
  (𝓝[>] 0) (𝓝 0)h:ℝ⊢ h⁻¹ ^ n * modifiedDifference h ((modifiedDifference h)^[k] f) (a - c * h) =
  h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c - 1) * h) + h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c + 1) * h) -
    2 * Real.cos (h / 2) * (h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h))
    unfold modifiedDifferencef:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)k:ℕih:∀ {c : ℝ}, ↑k < c → Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0)c:ℝhc:↑k + 1 < chp:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c - 1) * h)) (𝓝[>] 0) (𝓝 0)hm:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c + 1) * h)) (𝓝[>] 0) (𝓝 0)h0:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0)hcos:Tendsto (fun h => 2 * Real.cos (h / 2)) (𝓝[>] 0) (𝓝 2)hl:Tendsto
  (fun x =>
    x⁻¹ ^ n * (modifiedDifference x)^[k] f (a - (c - 1) * x) +
        x⁻¹ ^ n * (modifiedDifference x)^[k] f (a - (c + 1) * x) -
      2 * Real.cos (x / 2) * (x⁻¹ ^ n * (modifiedDifference x)^[k] f (a - c * x)))
  (𝓝[>] 0) (𝓝 0)h:ℝ⊢ h⁻¹ ^ n *
    ((fun f x => f (x + h) + f (x - h) - 2 * Real.cos (h / 2) * f x)^[k] f (a - c * h + h) +
        (fun f x => f (x + h) + f (x - h) - 2 * Real.cos (h / 2) * f x)^[k] f (a - c * h - h) -
      2 * Real.cos (h / 2) * (fun f x => f (x + h) + f (x - h) - 2 * Real.cos (h / 2) * f x)^[k] f (a - c * h)) =
  h⁻¹ ^ n * (fun f x => f (x + h) + f (x - h) - 2 * Real.cos (h / 2) * f x)^[k] f (a - (c - 1) * h) +
      h⁻¹ ^ n * (fun f x => f (x + h) + f (x - h) - 2 * Real.cos (h / 2) * f x)^[k] f (a - (c + 1) * h) -
    2 * Real.cos (h / 2) * (h⁻¹ ^ n * (fun f x => f (x + h) + f (x - h) - 2 * Real.cos (h / 2) * f x)^[k] f (a - c * h))
    have hp : a - c * h + h = a - (c - 1) * h := byf:ℝ → ℝa:ℝn:ℕk:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)c:ℝhc:↑k < c⊢ Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0) ringf:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)k:ℕih:∀ {c : ℝ}, ↑k < c → Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0)c:ℝhc:↑k + 1 < chp✝:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c - 1) * h)) (𝓝[>] 0) (𝓝 0)hm:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c + 1) * h)) (𝓝[>] 0) (𝓝 0)h0:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0)hcos:Tendsto (fun h => 2 * Real.cos (h / 2)) (𝓝[>] 0) (𝓝 2)hl:Tendsto
  (fun x =>
    x⁻¹ ^ n * (modifiedDifference x)^[k] f (a - (c - 1) * x) +
        x⁻¹ ^ n * (modifiedDifference x)^[k] f (a - (c + 1) * x) -
      2 * Real.cos (x / 2) * (x⁻¹ ^ n * (modifiedDifference x)^[k] f (a - c * x)))
  (𝓝[>] 0) (𝓝 0)h:ℝhp:a - c * h + h = a - (c - 1) * h⊢ h⁻¹ ^ n *
    ((fun f x => f (x + h) + f (x - h) - 2 * Real.cos (h / 2) * f x)^[k] f (a - c * h + h) +
        (fun f x => f (x + h) + f (x - h) - 2 * Real.cos (h / 2) * f x)^[k] f (a - c * h - h) -
      2 * Real.cos (h / 2) * (fun f x => f (x + h) + f (x - h) - 2 * Real.cos (h / 2) * f x)^[k] f (a - c * h)) =
  h⁻¹ ^ n * (fun f x => f (x + h) + f (x - h) - 2 * Real.cos (h / 2) * f x)^[k] f (a - (c - 1) * h) +
      h⁻¹ ^ n * (fun f x => f (x + h) + f (x - h) - 2 * Real.cos (h / 2) * f x)^[k] f (a - (c + 1) * h) -
    2 * Real.cos (h / 2) * (h⁻¹ ^ n * (fun f x => f (x + h) + f (x - h) - 2 * Real.cos (h / 2) * f x)^[k] f (a - c * h))
    have hm : a - c * h - h = a - (c + 1) * h := byf:ℝ → ℝa:ℝn:ℕk:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)c:ℝhc:↑k < c⊢ Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0) ringf:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)k:ℕih:∀ {c : ℝ}, ↑k < c → Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0)c:ℝhc:↑k + 1 < chp✝:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c - 1) * h)) (𝓝[>] 0) (𝓝 0)hm✝:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c + 1) * h)) (𝓝[>] 0) (𝓝 0)h0:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0)hcos:Tendsto (fun h => 2 * Real.cos (h / 2)) (𝓝[>] 0) (𝓝 2)hl:Tendsto
  (fun x =>
    x⁻¹ ^ n * (modifiedDifference x)^[k] f (a - (c - 1) * x) +
        x⁻¹ ^ n * (modifiedDifference x)^[k] f (a - (c + 1) * x) -
      2 * Real.cos (x / 2) * (x⁻¹ ^ n * (modifiedDifference x)^[k] f (a - c * x)))
  (𝓝[>] 0) (𝓝 0)h:ℝhp:a - c * h + h = a - (c - 1) * hhm:a - c * h - h = a - (c + 1) * h⊢ h⁻¹ ^ n *
    ((fun f x => f (x + h) + f (x - h) - 2 * Real.cos (h / 2) * f x)^[k] f (a - c * h + h) +
        (fun f x => f (x + h) + f (x - h) - 2 * Real.cos (h / 2) * f x)^[k] f (a - c * h - h) -
      2 * Real.cos (h / 2) * (fun f x => f (x + h) + f (x - h) - 2 * Real.cos (h / 2) * f x)^[k] f (a - c * h)) =
  h⁻¹ ^ n * (fun f x => f (x + h) + f (x - h) - 2 * Real.cos (h / 2) * f x)^[k] f (a - (c - 1) * h) +
      h⁻¹ ^ n * (fun f x => f (x + h) + f (x - h) - 2 * Real.cos (h / 2) * f x)^[k] f (a - (c + 1) * h) -
    2 * Real.cos (h / 2) * (h⁻¹ ^ n * (fun f x => f (x + h) + f (x - h) - 2 * Real.cos (h / 2) * f x)^[k] f (a - c * h))
    rw [hp,f:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)k:ℕih:∀ {c : ℝ}, ↑k < c → Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0)c:ℝhc:↑k + 1 < chp✝:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c - 1) * h)) (𝓝[>] 0) (𝓝 0)hm✝:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c + 1) * h)) (𝓝[>] 0) (𝓝 0)h0:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0)hcos:Tendsto (fun h => 2 * Real.cos (h / 2)) (𝓝[>] 0) (𝓝 2)hl:Tendsto
  (fun x =>
    x⁻¹ ^ n * (modifiedDifference x)^[k] f (a - (c - 1) * x) +
        x⁻¹ ^ n * (modifiedDifference x)^[k] f (a - (c + 1) * x) -
      2 * Real.cos (x / 2) * (x⁻¹ ^ n * (modifiedDifference x)^[k] f (a - c * x)))
  (𝓝[>] 0) (𝓝 0)h:ℝhp:a - c * h + h = a - (c - 1) * hhm:a - c * h - h = a - (c + 1) * h⊢ h⁻¹ ^ n *
    ((fun f x => f (x + h) + f (x - h) - 2 * Real.cos (h / 2) * f x)^[k] f (a - (c - 1) * h) +
        (fun f x => f (x + h) + f (x - h) - 2 * Real.cos (h / 2) * f x)^[k] f (a - c * h - h) -
      2 * Real.cos (h / 2) * (fun f x => f (x + h) + f (x - h) - 2 * Real.cos (h / 2) * f x)^[k] f (a - c * h)) =
  h⁻¹ ^ n * (fun f x => f (x + h) + f (x - h) - 2 * Real.cos (h / 2) * f x)^[k] f (a - (c - 1) * h) +
      h⁻¹ ^ n * (fun f x => f (x + h) + f (x - h) - 2 * Real.cos (h / 2) * f x)^[k] f (a - (c + 1) * h) -
    2 * Real.cos (h / 2) * (h⁻¹ ^ n * (fun f x => f (x + h) + f (x - h) - 2 * Real.cos (h / 2) * f x)^[k] f (a - c * h)) hmf:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)k:ℕih:∀ {c : ℝ}, ↑k < c → Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0)c:ℝhc:↑k + 1 < chp✝:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c - 1) * h)) (𝓝[>] 0) (𝓝 0)hm✝:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c + 1) * h)) (𝓝[>] 0) (𝓝 0)h0:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0)hcos:Tendsto (fun h => 2 * Real.cos (h / 2)) (𝓝[>] 0) (𝓝 2)hl:Tendsto
  (fun x =>
    x⁻¹ ^ n * (modifiedDifference x)^[k] f (a - (c - 1) * x) +
        x⁻¹ ^ n * (modifiedDifference x)^[k] f (a - (c + 1) * x) -
      2 * Real.cos (x / 2) * (x⁻¹ ^ n * (modifiedDifference x)^[k] f (a - c * x)))
  (𝓝[>] 0) (𝓝 0)h:ℝhp:a - c * h + h = a - (c - 1) * hhm:a - c * h - h = a - (c + 1) * h⊢ h⁻¹ ^ n *
    ((fun f x => f (x + h) + f (x - h) - 2 * Real.cos (h / 2) * f x)^[k] f (a - (c - 1) * h) +
        (fun f x => f (x + h) + f (x - h) - 2 * Real.cos (h / 2) * f x)^[k] f (a - (c + 1) * h) -
      2 * Real.cos (h / 2) * (fun f x => f (x + h) + f (x - h) - 2 * Real.cos (h / 2) * f x)^[k] f (a - c * h)) =
  h⁻¹ ^ n * (fun f x => f (x + h) + f (x - h) - 2 * Real.cos (h / 2) * f x)^[k] f (a - (c - 1) * h) +
      h⁻¹ ^ n * (fun f x => f (x + h) + f (x - h) - 2 * Real.cos (h / 2) * f x)^[k] f (a - (c + 1) * h) -
    2 * Real.cos (h / 2) * (h⁻¹ ^ n * (fun f x => f (x + h) + f (x - h) - 2 * Real.cos (h / 2) * f x)^[k] f (a - c * h))]f:ℝ → ℝa:ℝn:ℕhf:Tendsto (fun u => u⁻¹ ^ n * f (a - u)) (𝓝[>] 0) (𝓝 0)k:ℕih:∀ {c : ℝ}, ↑k < c → Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0)c:ℝhc:↑k + 1 < chp✝:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c - 1) * h)) (𝓝[>] 0) (𝓝 0)hm✝:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - (c + 1) * h)) (𝓝[>] 0) (𝓝 0)h0:Tendsto (fun h => h⁻¹ ^ n * (modifiedDifference h)^[k] f (a - c * h)) (𝓝[>] 0) (𝓝 0)hcos:Tendsto (fun h => 2 * Real.cos (h / 2)) (𝓝[>] 0) (𝓝 2)hl:Tendsto
  (fun x =>
    x⁻¹ ^ n * (modifiedDifference x)^[k] f (a - (c - 1) * x) +
        x⁻¹ ^ n * (modifiedDifference x)^[k] f (a - (c + 1) * x) -
      2 * Real.cos (x / 2) * (x⁻¹ ^ n * (modifiedDifference x)^[k] f (a - c * x)))
  (𝓝[>] 0) (𝓝 0)h:ℝhp:a - c * h + h = a - (c - 1) * hhm:a - c * h - h = a - (c + 1) * h⊢ h⁻¹ ^ n *
    ((fun f x => f (x + h) + f (x - h) - 2 * Real.cos (h / 2) * f x)^[k] f (a - (c - 1) * h) +
        (fun f x => f (x + h) + f (x - h) - 2 * Real.cos (h / 2) * f x)^[k] f (a - (c + 1) * h) -
      2 * Real.cos (h / 2) * (fun f x => f (x + h) + f (x - h) - 2 * Real.cos (h / 2) * f x)^[k] f (a - c * h)) =
  h⁻¹ ^ n * (fun f x => f (x + h) + f (x - h) - 2 * Real.cos (h / 2) * f x)^[k] f (a - (c - 1) * h) +
      h⁻¹ ^ n * (fun f x => f (x + h) + f (x - h) - 2 * Real.cos (h / 2) * f x)^[k] f (a - (c + 1) * h) -
    2 * Real.cos (h / 2) * (h⁻¹ ^ n * (fun f x => f (x + h) + f (x - h) - 2 * Real.cos (h / 2) * f x)^[k] f (a - c * h))
    ringAll goals completed! 🐙

theorem tendsto_thetaBoundaryRemainder_re_weighted (n : ℕ) :
    Tendsto (fun u : ℝ ↦ u⁻¹ ^ n * (thetaBoundaryRemainder (Real.pi / 4 - u)).re)
      (𝓝[>] 0) (𝓝 0) := byn:ℕ⊢ Tendsto (fun u => u⁻¹ ^ n * (thetaBoundaryRemainder (Real.pi / 4 - u)).re) (𝓝[>] 0) (𝓝 0)
  apply squeeze_zero_norm' _ (tendsto_thetaBoundaryRemainder_weighted n)n:ℕ⊢ ∀ᶠ (n_1 : ℝ) in 𝓝[>] 0,
  ‖n_1⁻¹ ^ n * (thetaBoundaryRemainder (Real.pi / 4 - n_1)).re‖ ≤
    n_1⁻¹ ^ n * ‖thetaBoundaryRemainder (Real.pi / 4 - n_1)‖
  filter_upwards [self_mem_nhdsWithin] with u hun:ℕu:ℝhu:u ∈ Ioi 0⊢ ‖u⁻¹ ^ n * (thetaBoundaryRemainder (Real.pi / 4 - u)).re‖ ≤ u⁻¹ ^ n * ‖thetaBoundaryRemainder (Real.pi / 4 - u)‖
  rw [norm_mul,n:ℕu:ℝhu:u ∈ Ioi 0⊢ ‖u⁻¹ ^ n‖ * ‖(thetaBoundaryRemainder (Real.pi / 4 - u)).re‖ ≤ u⁻¹ ^ n * ‖thetaBoundaryRemainder (Real.pi / 4 - u)‖ Real.norm_eq_abs,n:ℕu:ℝhu:u ∈ Ioi 0⊢ |u⁻¹ ^ n| * ‖(thetaBoundaryRemainder (Real.pi / 4 - u)).re‖ ≤ u⁻¹ ^ n * ‖thetaBoundaryRemainder (Real.pi / 4 - u)‖ abs_of_pos (pow_pos (inv_pos.mpr hu) n)n:ℕu:ℝhu:u ∈ Ioi 0⊢ u⁻¹ ^ n * ‖(thetaBoundaryRemainder (Real.pi / 4 - u)).re‖ ≤ u⁻¹ ^ n * ‖thetaBoundaryRemainder (Real.pi / 4 - u)‖]n:ℕu:ℝhu:u ∈ Ioi 0⊢ u⁻¹ ^ n * ‖(thetaBoundaryRemainder (Real.pi / 4 - u)).re‖ ≤ u⁻¹ ^ n * ‖thetaBoundaryRemainder (Real.pi / 4 - u)‖
  exact mul_le_mul_of_nonneg_left (Complex.abs_re_le_norm _)
    (pow_nonneg (inv_nonneg.mpr (le_of_lt (show 0 < u from hu))) n)All goals completed! 🐙

theorem tendsto_scaled_theta_difference (k : ℕ) :
    Tendsto (fun h : ℝ ↦ h⁻¹ ^ (2 * k) *
      (modifiedDifference h)^[k] (fun α ↦ (thetaBoundaryRemainder α).re)
        (Real.pi / 4 - ((k : ℝ) + 1) * h)) (𝓝[>] 0) (𝓝 0) :=
  tendsto_scaled_iterate_modifiedDifference (2 * k) k
    (tendsto_thetaBoundaryRemainder_re_weighted (2 * k)) (byk:ℕ⊢ ↑k < ↑k + 1 linarithAll goals completed! 🐙)