theorem sum_variance_truncation_le {X : Ω → ℝ} (hint : Integrable X) (hnonneg : 0 ≤ X) (K : ℕ) :
    ∑ j in range K, ((j : ℝ) ^ 2)⁻¹ * 𝔼[truncation X j ^ 2] ≤ 2 * 𝔼[X] := by
  set Y := fun n : ℕ => truncation X n
  let ρ : Measure ℝ := Measure.map X ℙ
  have Y2 : ∀ n, 𝔼[Y n ^ 2] = ∫ x in (0)..n, x ^ 2 ∂ρ := by
    intro n
    change 𝔼[fun x => Y n x ^ 2] = _
    rw [moment_truncation_eq_intervalIntegral_of_nonneg hint.1 two_ne_zero hnonneg]
  

Proof state:
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : Ω → ℝ
hint : Integrable X
hnonneg : 0 ≤ X
K : ℕ
Y : ℕ → Ω → ℝ := fun n => truncation X ↑n
ρ : Measure ℝ := Measure.map X ℙ
Y2 : ∀ (n : ℕ), ∫ (a : Ω), (Y n ^ 2) a = ∫ (x : ℝ) in 0 ..↑n, x ^ 2 ∂ρ
⊢ ∑ j in range K, (↑j ^ 2)⁻¹ * ∫ (a : Ω), (truncation X ↑j ^ 2) a ≤ 2 * ∫ (a : Ω), X a

Next tactic:
calc
    ∑ j in range K, ((j : ℝ) ^ 2)⁻¹ * 𝔼[Y j ^ 2] =
        ∑ j in range K, ((j : ℝ) ^ 2)⁻¹ * ∫ x in (0)..j, x ^ 2 ∂ρ := by simp_rw [Y2]
    _ = ∑ j in range K, ((j : ℝ) ^ 2)⁻¹ * ∑ k in range j, ∫ x in k..(k + 1 : ℕ), x ^ 2 ∂ρ := by
      congr 1 with j
      congr 1
      rw [intervalIntegral.sum_integral_adjacent_intervals]
      · norm_cast
      intro k _
      exact (continuous_id.pow _).intervalIntegrable _ _
    _ = ∑ k in range K, (∑ j in Ioo k K, ((j : ℝ) ^ 2)⁻¹) * ∫ x in k..(k + 1 : ℕ), x ^ 2 ∂ρ := by
      simp_rw [mul_sum, sum_mul, sum_sigma']
      refine' sum_bij' (fun (p : Σ _ : ℕ, ℕ) _ => (⟨p.2, p.1⟩ : Σ _ : ℕ, ℕ)) _ (fun a _ => rfl)
        (fun (p : Σ _ : ℕ, ℕ) _ => (⟨p.2, p.1⟩ : Σ _ : ℕ, ℕ)) _ _ _
      · rintro ⟨i, j⟩ hij
        simp only [mem_sigma, mem_range, mem_filter] at hij
        simp [hij, mem_sigma, mem_range, and_self_iff, hij.2.trans hij.1]
      · rintro ⟨i, j⟩ hij
        simp only [mem_sigma, mem_range, mem_Ioo] at hij
        simp only [hij, mem_sigma, mem_range, and_self_iff]
      · rintro ⟨i, j⟩ hij; rfl
      · rintro ⟨i, j⟩ hij; rfl
    _ ≤ ∑ k in range K, 2 / (k + 1 : ℝ) * ∫ x in k..(k + 1 : ℕ), x ^ 2 ∂ρ := by
      apply sum_le_sum fun k _ => ?_
      refine' mul_le_mul_of_nonneg_right (sum_Ioo_inv_sq_le _ _) _
      refine' intervalIntegral.integral_nonneg_of_forall _ fun u => sq_nonneg _
      simp only [Nat.cast_add, Nat.cast_one, le_add_iff_nonneg_right, zero_le_one]
    _ ≤ ∑ k in range K, ∫ x in k..(k + 1 : ℕ), 2 * x ∂ρ := by
      apply sum_le_sum fun k _ => ?_
      have Ik : (k : ℝ) ≤ (k + 1 : ℕ) := by simp
      rw [← intervalIntegral.integral_const_mul, intervalIntegral.integral_of_le Ik,
        intervalIntegral.integral_of_le Ik]
      refine' set_integral_mono_on _ _ measurableSet_Ioc fun x hx => _
      · apply Continuous.integrableOn_Ioc
        exact continuous_const.mul (continuous_pow 2)
      · apply Continuous.integrableOn_Ioc
        exact continuous_const.mul continuous_id'
      · calc
          ↑2 / (↑k + ↑1) * x ^ 2 = x / (k + 1) * (2 * x) := by ring
          _ ≤ 1 * (2 * x) :=
            (mul_le_mul_of_nonneg_right (by
              convert (div_le_one _).2 hx.2; norm_cast
              simp only [Nat.cast_add, Nat.cast_one]
              linarith only [show (0 : ℝ) ≤ k from Nat.cast_nonneg k])
              (mul_nonneg zero_le_two ((Nat.cast_nonneg k).trans hx.1.le)))
          _ = 2 * x := by rw [one_mul]
    _ = 2 * ∫ x in (0 : ℝ)..K, x ∂ρ := by
      rw [intervalIntegral.sum_integral_adjacent_intervals fun k _ => ?_]
      swap; · exact (continuous_const.mul continuous_id').intervalIntegrable _ _
      rw [intervalIntegral.integral_const_mul]
      norm_cast
    _ ≤ 2 * 𝔼[X] := mul_le_mul_of_nonneg_left (by
      rw [← integral_truncation_eq_intervalIntegral_of_nonneg hint.1 hnonneg]
      exact integral_truncation_le_integral_of_nonneg hint hnonneg) zero_le_two