putnam-1965-b4

import Mathlib

open EuclideanGeometry Topology Filter Complex
noncomputable abbrev putnam_1965_b4_solution : ((((ℝ → ℝ) → (ℝ → ℝ)) × ((ℝ → ℝ) → (ℝ → ℝ))) × ((Set ℝ) × (ℝ → ℝ))) := ((fun h : ℝ → ℝ => h + (fun x : ℝ => x), fun h : ℝ → ℝ => h + (fun _ : ℝ => 1)), ({x : ℝ | x ≥ 0}, Real.sqrt))

theorem putnam_1965_b4 (f u v : ℕ → ℝ → ℝ)
    (hu : ∀ n > 0, ∀ x, u n x = ∑ i ∈ Finset.Icc 0 (n / 2), (n.choose (2 * i)) * x ^ i)
    (hv : ∀ n > 0, ∀ x, v n x = ∑ i ∈ Finset.Icc 0 ((n - 1) / 2), (n.choose (2 * i + 1)) * x ^ i)
    (hf : ∀ n > 0, ∀ x, f n x = u n x / v n x)
    (n : ℕ)
    (hn : 0 < n) :
    let ⟨⟨p, q⟩, ⟨s, g⟩⟩ := putnam_1965_b4_solution
    (∀ x, v n x ≠ 0 → v (n + 1) x ≠ 0 → q (f n) x ≠ 0 → f (n + 1) x = p (f n) x / q (f n) x) ∧
    s = {x | ∃ l, Tendsto (fun n ↦ f n x) atTop (𝓝 l)} ∧
    ∀ x ∈ s, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x)) := by
  sorry