. refine Continuous.tendsto' ?_ _ _ (by simp)
  refine Continuous.abs ?_
  exact Continuous.sub hφ.continuous continuous_const
case hδ
φ : ℝ → ℝ
hφ : IsBump φ
h_integrable : ∀ {n : ℕ+}, Integrable fun x => (φ x - φ 0) * DiracSeq (↑↑n) x
⊢ Tendsto (fun x => |φ x - φ 0|) (nhds 0) (nhds 0)

case h_tendsto
φ : ℝ → ℝ
hφ : IsBump φ
h_integrable : ∀ {n : ℕ+}, Integrable fun x => (φ x - φ 0) * DiracSeq (↑↑n) x
hδ : Tendsto (fun x => |φ x - φ 0|) (nhds 0) (nhds 0)
⊢ Tendsto (fun n => ∫ (x : ℝ), (φ x - φ 0) * DiracSeq (↑↑n) x) atTop (nhds 0)
case h_tendsto
φ : ℝ → ℝ
hφ : IsBump φ
h_integrable : ∀ {n : ℕ+}, Integrable fun x => (φ x - φ 0) * DiracSeq (↑↑n) x
hδ : Tendsto (fun x => |φ x - φ 0|) (nhds 0) (nhds 0)
⊢ Tendsto (fun n => ∫ (x : ℝ), (φ x - φ 0) * DiracSeq (↑↑n) x) atTop (nhds 0)