have l4 : 0 < ∫ (y : ℝ) in Ioi 0, ψ y := by
  have r1 : 0 ≤ᵐ[Measure.restrict volume (Ioi 0)] ψ := eventually_of_forall l2
  have r2 : IntegrableOn (fun y ↦ ψ y) (Ioi 0) volume :=
    (h1.continuous.integrable_of_hasCompactSupport h2).integrableOn
  have r3 : Ico 1 2 ⊆ Function.support ψ := by intro x hx ; have := h4 x ; simp [hx] at this ⊢ ; linarith
  have r4 : Ico 1 2 ⊆ Function.support ψ ∩ Ioi 0 := by
    simp only [subset_inter_iff, r3, true_and] ; apply Ico_subset_Icc_self.trans ; rw [Icc_subset_Ioi_iff] <;> linarith
  have r5 : 1 ≤ volume ((Function.support fun y ↦ ψ y) ∩ Ioi 0) := by convert volume.mono r4 ; norm_num
  simpa [setIntegral_pos_iff_support_of_nonneg_ae r1 r2] using zero_lt_one.trans_le r5
case intro.intro.intro.intro.intro.intro
n : ℕ
A a b c d u x y t σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f✝ : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
f : ℕ → ℝ
hpos : 0 ≤ f
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')
hcheby : cheby fun n => ↑(f n)
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}
S : (ℝ → ℝ) → ℝ → ℝ := fun g x => (∑' (n : ℕ), f n * g (↑n / x)) / x
hSnonneg : ∀ {g : ℝ → ℝ}, 0 ≤ g → ∀ᶠ (x : ℝ) in atTop, 0 ≤ S g x
ε : ℝ
ψ : ℝ → ℝ
h1 : ContDiff ℝ ⊤ ψ
h2 : HasCompactSupport ψ
h3 : closure (Function.support ψ) ⊆ Ioi 0
h4 : (Ico 1 2).indicator 1 ≤ ψ
key : Tendsto (fun x => (∑' (n : ℕ), f n * ψ (↑n / x)) / x) atTop (𝓝 (A * ∫ (y : ℝ) in Ioi 0, ψ y))
l2 : 0 ≤ ψ
l1 : ∀ᶠ (x : ℝ) in atTop, 0 ≤ S ψ x
l3 : 0 ≤ A * ∫ (y : ℝ) in Ioi 0, ψ y
⊢ 0 ≤ A
case intro.intro.intro.intro.intro.intro
n : ℕ
A a b c d u x y t σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f✝ : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
f : ℕ → ℝ
hpos : 0 ≤ f
hf : ∀ (σ' : ℝ), 1 < σ' → Summable (nterm (fun n => ↑(f n)) σ')
hcheby : cheby fun n => ↑(f n)
hG : ContinuousOn G {s | 1 ≤ s.re}
hG' : EqOn G (fun s => LSeries (fun n => ↑(f n)) s - ↑A / (s - 1)) {s | 1 < s.re}
S : (ℝ → ℝ) → ℝ → ℝ := fun g x => (∑' (n : ℕ), f n * g (↑n / x)) / x
hSnonneg : ∀ {g : ℝ → ℝ}, 0 ≤ g → ∀ᶠ (x : ℝ) in atTop, 0 ≤ S g x
ε : ℝ
ψ : ℝ → ℝ
h1 : ContDiff ℝ ⊤ ψ
h2 : HasCompactSupport ψ
h3 : closure (Function.support ψ) ⊆ Ioi 0
h4 : (Ico 1 2).indicator 1 ≤ ψ
key : Tendsto (fun x => (∑' (n : ℕ), f n * ψ (↑n / x)) / x) atTop (𝓝 (A * ∫ (y : ℝ) in Ioi 0, ψ y))
l2 : 0 ≤ ψ
l1 : ∀ᶠ (x : ℝ) in atTop, 0 ≤ S ψ x
l3 : 0 ≤ A * ∫ (y : ℝ) in Ioi 0, ψ y
l4 : 0 < ∫ (y : ℝ) in Ioi 0, ψ y
⊢ 0 ≤ A