-- Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal
Goal:
F : Type u_3
inst : NormedAddCommGroup F
inst_1 : InnerProductSpace ℝ F
K : Set F
ne : K.Nonempty
h₁ : IsComplete K
h₂ : Convex ℝ K
u : F
δ : ℝ := ⨅ (w : (↑K : Type u_3)), ‖u - (↑w : F)‖
this : Nonempty.{u_3 + 1} (↑K : Type u_3) := ⋯
zero_le_δ : 0 ≤ δ
δ_le : ∀ (w : (↑K : Type u_3)), δ ≤ ‖u - (↑w : F)‖
δ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖
w : ℕ → (↑K : Type u_3)
hw : ∀ (n : ℕ), ‖u - (↑(w n) : F)‖ < δ + 1 / HAdd.hAdd (α := ℝ) (↑n : ℝ) 1
norm_tendsto : Tendsto (fun (n : ℕ) ↦ ‖u - (↑(w n) : F)‖) atTop (𝓝 δ)
⊢ CauchySeq.{u_3, 0} (α := F) fun (n : ℕ) ↦ (↑(w n) : F)

Tactic:
have seq_is_cauchy : CauchySeq fun n => (w n : F) := by
    rw [cauchySeq_iff_le_tendsto_0]
    -- splits into three goals
    let b := fun n : ℕ => 8 * δ * (1 / (n + 1)) + 4 * (1 / (n + 1)) * (1 / (n + 1))
    use fun n => √(b n)
    constructor
    -- first goal :  `∀ (n : ℕ), 0 ≤ √(b n)`
    · intro n
      exact sqrt_nonneg _
    constructor
    -- second goal : `∀ (n m N : ℕ), N ≤ n → N ≤ m → dist ↑(w n) ↑(w m) ≤ √(b N)`
    · intro p q N hp hq
      let wp := (w p : F)
      let wq := (w q : F)
      let a := u - wq
      let b := u - wp
      let half := 1 / (2 : ℝ)
      let div := 1 / ((N : ℝ) + 1)
      have :
        4 * ‖u - half • (wq + wp)‖ * ‖u - half • (wq + wp)‖ + ‖wp - wq‖ * ‖wp - wq‖ =
          2 * (‖a‖ * ‖a‖ + ‖b‖ * ‖b‖) :=
        calc
          4 * ‖u - half • (wq + wp)‖ * ‖u - half • (wq + wp)‖ + ‖wp - wq‖ * ‖wp - wq‖ =
              2 * ‖u - half • (wq + wp)‖ * (2 * ‖u - half • (wq + wp)‖) + ‖wp - wq‖ * ‖wp - wq‖ :=
            by ring
          _ =
              absR 2 * ‖u - half • (wq + wp)‖ * (absR 2 * ‖u - half • (wq + wp)‖) +
                ‖wp - wq‖ * ‖wp - wq‖ := by
            rw [abs_of_nonneg]
            exact zero_le_two
          _ =
              ‖(2 : ℝ) • (u - half • (wq + wp))‖ * ‖(2 : ℝ) • (u - half • (wq + wp))‖ +
                ‖wp - wq‖ * ‖wp - wq‖ := by simp [norm_smul]
          _ = ‖a + b‖ * ‖a + b‖ + ‖a - b‖ * ‖a - b‖ := by
            rw [smul_sub, smul_smul, mul_one_div_cancel (_root_.two_ne_zero : (2 : ℝ) ≠ 0), ←
              one_add_one_eq_two, add_smul]
            simp only [one_smul]
            have eq₁ : wp - wq = a - b := (sub_sub_sub_cancel_left _ _ _).symm
            have eq₂ : u + u - (wq + wp) = a + b := by
              change u + u - (wq + wp) = u - wq + (u - wp)
              abel
            rw [eq₁, eq₂]
          _ = 2 * (‖a‖ * ‖a‖ + ‖b‖ * ‖b‖) := parallelogram_law_with_norm_mul ℝ _ _
      have eq : δ ≤ ‖u - half • (wq + wp)‖ := by
        rw [smul_add]
        apply δ_le'
        apply h₂
        repeat' exact Subtype.mem _
        repeat' exact le_of_lt one_half_pos
        exact add_halves 1
      have eq₂ : ‖a‖ ≤ δ + div := by grw [hw, Nat.one_div_le_one_div hq]
      have eq₂' : ‖b‖ ≤ δ + div := by grw [hw, Nat.one_div_le_one_div hp]
      rw [dist_eq_norm]
      apply nonneg_le_nonneg_of_sq_le_sq
      · exact sqrt_nonneg _
      rw [mul_self_sqrt]
      · calc
        ‖wp - wq‖ * ‖wp - wq‖ =
            2 * (‖a‖ * ‖a‖ + ‖b‖ * ‖b‖) - 4 * ‖u - half • (wq + wp)‖ * ‖u - half • (wq + wp)‖ := by
          simp [← this]
        _ ≤ 2 * (‖a‖ * ‖a‖ + ‖b‖ * ‖b‖) - 4 * δ * δ := by gcongr
        _ ≤ 2 * ((δ + div) * (δ + div) + (δ + div) * (δ + div)) - 4 * δ * δ := by gcongr
        _ = 8 * δ * div + 4 * div * div := by ring
      positivity
    -- third goal : `Tendsto (fun (n : ℕ) => √(b n)) atTop (𝓝 0)`
    suffices Tendsto (fun x ↦ √(8 * δ * x + 4 * x * x) : ℝ → ℝ) (𝓝 0) (𝓝 0)
      from this.comp tendsto_one_div_add_atTop_nhds_zero_nat
    exact Continuous.tendsto' (by fun_prop) _ _ (by simp)