-- Mathlib/MeasureTheory/Measure/SeparableMeasure.lean
theorem Measure.MeasureDense.of_generateFrom_isSetAlgebra_finite [IsFiniteMeasure μ]
    (h𝒜 : IsSetAlgebra 𝒜) (hgen : m = MeasurableSpace.generateFrom 𝒜) : μ.MeasureDense 𝒜 where
  measurable s hs := hgen ▸ measurableSet_generateFrom hs
  approx s ms

Proof state:
case h₁
X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
inst✝ : IsFiniteMeasure μ
h𝒜 : IsSetAlgebra 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
ms : MeasurableSet s
x✝ : Set X
f : ℕ → Set X
hs✝ : ∀ (n : ℕ), MeasurableSet (f n)
hf : ∀ (n : ℕ), MeasurableSet (f n) ∧ ∀ (ε : ℝ), 0 < ε → ∃ t ∈ 𝒜, (μ (f n ∆ t)).toReal < ε
ε : ℝ
ε_pos : 0 < ε
this : Filter.Tendsto (fun n => (μ (Accumulate f n)).toReal) Filter.atTop (nhds (μ (⋃ i, f i)).toReal)
N : ℕ
hN : ∀ n ≥ N, dist (μ (Accumulate f n)).toReal (μ (⋃ i, f i)).toReal < ε / 2
g : ℕ → Set X
g_mem : ∀ (n : ℕ), g n ∈ 𝒜
hg : ∀ (n : ℕ), (μ (f n ∆ g n)).toReal < ε / (2 * (↑N + 1))
⊢ (μ ((⋃ n, f n) \ ⋃ n ∈ Finset.range (N + 1), f n)).toReal < ε / 2

Tactic:
rw [measure_diff (h_fin := measure_ne_top _ _),
  toReal_sub_of_le (ha := measure_ne_top _ _)]

Result:
case h₁
X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
inst✝ : IsFiniteMeasure μ
h𝒜 : IsSetAlgebra 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
ms : MeasurableSet s
x✝ : Set X
f : ℕ → Set X
hs✝ : ∀ (n : ℕ), MeasurableSet (f n)
hf : ∀ (n : ℕ), MeasurableSet (f n) ∧ ∀ (ε : ℝ), 0 < ε → ∃ t ∈ 𝒜, (μ (f n ∆ t)).toReal < ε
ε : ℝ
ε_pos : 0 < ε
this : Filter.Tendsto (fun n => (μ (Accumulate f n)).toReal) Filter.atTop (nhds (μ (⋃ i, f i)).toReal)
N : ℕ
hN : ∀ n ≥ N, dist (μ (Accumulate f n)).toReal (μ (⋃ i, f i)).toReal < ε / 2
g : ℕ → Set X
g_mem : ∀ (n : ℕ), g n ∈ 𝒜
hg : ∀ (n : ℕ), (μ (f n ∆ g n)).toReal < ε / (2 * (↑N + 1))
⊢ (μ (⋃ n, f n)).toReal - (μ (⋃ n ∈ Finset.range (N + 1), f n)).toReal < ε / 2

case h₁
X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
inst✝ : IsFiniteMeasure μ
h𝒜 : IsSetAlgebra 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
ms : MeasurableSet s
x✝ : Set X
f : ℕ → Set X
hs✝ : ∀ (n : ℕ), MeasurableSet (f n)
hf : ∀ (n : ℕ), MeasurableSet (f n) ∧ ∀ (ε : ℝ), 0 < ε → ∃ t ∈ 𝒜, (μ (f n ∆ t)).toReal < ε
ε : ℝ
ε_pos : 0 < ε
this : Filter.Tendsto (fun n => (μ (Accumulate f n)).toReal) Filter.atTop (nhds (μ (⋃ i, f i)).toReal)
N : ℕ
hN : ∀ n ≥ N, dist (μ (Accumulate f n)).toReal (μ (⋃ i, f i)).toReal < ε / 2
g : ℕ → Set X
g_mem : ∀ (n : ℕ), g n ∈ 𝒜
hg : ∀ (n : ℕ), (μ (f n ∆ g n)).toReal < ε / (2 * (↑N + 1))
⊢ μ (⋃ n ∈ Finset.range (N + 1), f n) ≤ μ (⋃ n, f n)

case h₁.h
X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
inst✝ : IsFiniteMeasure μ
h𝒜 : IsSetAlgebra 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
ms : MeasurableSet s
x✝ : Set X
f : ℕ → Set X
hs✝ : ∀ (n : ℕ), MeasurableSet (f n)
hf : ∀ (n : ℕ), MeasurableSet (f n) ∧ ∀ (ε : ℝ), 0 < ε → ∃ t ∈ 𝒜, (μ (f n ∆ t)).toReal < ε
ε : ℝ
ε_pos : 0 < ε
this : Filter.Tendsto (fun n => (μ (Accumulate f n)).toReal) Filter.atTop (nhds (μ (⋃ i, f i)).toReal)
N : ℕ
hN : ∀ n ≥ N, dist (μ (Accumulate f n)).toReal (μ (⋃ i, f i)).toReal < ε / 2
g : ℕ → Set X
g_mem : ∀ (n : ℕ), g n ∈ 𝒜
hg : ∀ (n : ℕ), (μ (f n ∆ g n)).toReal < ε / (2 * (↑N + 1))
⊢ ⋃ n ∈ Finset.range (N + 1), f n ⊆ ⋃ n, f n

case h₁.h₂
X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
inst✝ : IsFiniteMeasure μ
h𝒜 : IsSetAlgebra 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
ms : MeasurableSet s
x✝ : Set X
f : ℕ → Set X
hs✝ : ∀ (n : ℕ), MeasurableSet (f n)
hf : ∀ (n : ℕ), MeasurableSet (f n) ∧ ∀ (ε : ℝ), 0 < ε → ∃ t ∈ 𝒜, (μ (f n ∆ t)).toReal < ε
ε : ℝ
ε_pos : 0 < ε
this : Filter.Tendsto (fun n => (μ (Accumulate f n)).toReal) Filter.atTop (nhds (μ (⋃ i, f i)).toReal)
N : ℕ
hN : ∀ n ≥ N, dist (μ (Accumulate f n)).toReal (μ (⋃ i, f i)).toReal < ε / 2
g : ℕ → Set X
g_mem : ∀ (n : ℕ), g n ∈ 𝒜
hg : ∀ (n : ℕ), (μ (f n ∆ g n)).toReal < ε / (2 * (↑N + 1))
⊢ NullMeasurableSet (⋃ n ∈ Finset.range (N + 1), f n) μ

Proof state:
case h₁
X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
inst✝ : IsFiniteMeasure μ
h𝒜 : IsSetAlgebra 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
ms : MeasurableSet s
x✝ : Set X
f : ℕ → Set X
hs✝ : ∀ (n : ℕ), MeasurableSet (f n)
hf : ∀ (n : ℕ), MeasurableSet (f n) ∧ ∀ (ε : ℝ), 0 < ε → ∃ t ∈ 𝒜, (μ (f n ∆ t)).toReal < ε
ε : ℝ
ε_pos : 0 < ε
this : Filter.Tendsto (fun n => (μ (Accumulate f n)).toReal) Filter.atTop (nhds (μ (⋃ i, f i)).toReal)
N : ℕ
hN : ∀ n ≥ N, dist (μ (Accumulate f n)).toReal (μ (⋃ i, f i)).toReal < ε / 2
g : ℕ → Set X
g_mem : ∀ (n : ℕ), g n ∈ 𝒜
hg : ∀ (n : ℕ), (μ (f n ∆ g n)).toReal < ε / (2 * (↑N + 1))
⊢ (μ (⋃ n, f n)).toReal - (μ (⋃ n ∈ Finset.range (N + 1), f n)).toReal < ε / 2

Tactic:
apply lt_of_le_of_lt (sub_le_dist ..)

Result:
case h₁
X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
inst✝ : IsFiniteMeasure μ
h𝒜 : IsSetAlgebra 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
ms : MeasurableSet s
x✝ : Set X
f : ℕ → Set X
hs✝ : ∀ (n : ℕ), MeasurableSet (f n)
hf : ∀ (n : ℕ), MeasurableSet (f n) ∧ ∀ (ε : ℝ), 0 < ε → ∃ t ∈ 𝒜, (μ (f n ∆ t)).toReal < ε
ε : ℝ
ε_pos : 0 < ε
this : Filter.Tendsto (fun n => (μ (Accumulate f n)).toReal) Filter.atTop (nhds (μ (⋃ i, f i)).toReal)
N : ℕ
hN : ∀ n ≥ N, dist (μ (Accumulate f n)).toReal (μ (⋃ i, f i)).toReal < ε / 2
g : ℕ → Set X
g_mem : ∀ (n : ℕ), g n ∈ 𝒜
hg : ∀ (n : ℕ), (μ (f n ∆ g n)).toReal < ε / (2 * (↑N + 1))
⊢ dist (μ (⋃ n, f n)).toReal (μ (⋃ n ∈ Finset.range (N + 1), f n)).toReal < ε / 2

Proof state:
X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
inst✝ : IsFiniteMeasure μ
h𝒜 : IsSetAlgebra 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
ms : MeasurableSet s
x✝ : Set X
f : ℕ → Set X
hs✝ : ∀ (n : ℕ), MeasurableSet (f n)
hf : ∀ (n : ℕ), MeasurableSet (f n) ∧ ∀ (ε : ℝ), 0 < ε → ∃ t ∈ 𝒜, (μ (f n ∆ t)).toReal < ε
ε : ℝ
ε_pos : 0 < ε
this : Filter.Tendsto (fun n => (μ (Accumulate f n)).toReal) Filter.atTop (nhds (μ (⋃ i, f i)).toReal)
N : ℕ
hN : ∀ n ≥ N, dist (μ (Accumulate f n)).toReal (μ (⋃ i, f i)).toReal < ε / 2
g : ℕ → Set X
g_mem : ∀ (n : ℕ), g n ∈ 𝒜
hg : ∀ (n : ℕ), (μ (f n ∆ g n)).toReal < ε / (2 * (↑N + 1))
⊢ (μ ((⋃ n, f n) \ ⋃ n ∈ Finset.range (N + 1), f n)).toReal +
      (μ ((⋃ n ∈ Finset.range (N + 1), f n) ∆ ⋃ n ∈ Finset.range (N + 1), g n)).toReal <
    ε / 2 + ε / 2

Tactic:
apply _root_.add_lt_add

Result:
case h₁
X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
inst✝ : IsFiniteMeasure μ
h𝒜 : IsSetAlgebra 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
ms : MeasurableSet s
x✝ : Set X
f : ℕ → Set X
hs✝ : ∀ (n : ℕ), MeasurableSet (f n)
hf : ∀ (n : ℕ), MeasurableSet (f n) ∧ ∀ (ε : ℝ), 0 < ε → ∃ t ∈ 𝒜, (μ (f n ∆ t)).toReal < ε
ε : ℝ
ε_pos : 0 < ε
this : Filter.Tendsto (fun n => (μ (Accumulate f n)).toReal) Filter.atTop (nhds (μ (⋃ i, f i)).toReal)
N : ℕ
hN : ∀ n ≥ N, dist (μ (Accumulate f n)).toReal (μ (⋃ i, f i)).toReal < ε / 2
g : ℕ → Set X
g_mem : ∀ (n : ℕ), g n ∈ 𝒜
hg : ∀ (n : ℕ), (μ (f n ∆ g n)).toReal < ε / (2 * (↑N + 1))
⊢ (μ ((⋃ n, f n) \ ⋃ n ∈ Finset.range (N + 1), f n)).toReal < ε / 2

case h₂
X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
inst✝ : IsFiniteMeasure μ
h𝒜 : IsSetAlgebra 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
ms : MeasurableSet s
x✝ : Set X
f : ℕ → Set X
hs✝ : ∀ (n : ℕ), MeasurableSet (f n)
hf : ∀ (n : ℕ), MeasurableSet (f n) ∧ ∀ (ε : ℝ), 0 < ε → ∃ t ∈ 𝒜, (μ (f n ∆ t)).toReal < ε
ε : ℝ
ε_pos : 0 < ε
this : Filter.Tendsto (fun n => (μ (Accumulate f n)).toReal) Filter.atTop (nhds (μ (⋃ i, f i)).toReal)
N : ℕ
hN : ∀ n ≥ N, dist (μ (Accumulate f n)).toReal (μ (⋃ i, f i)).toReal < ε / 2
g : ℕ → Set X
g_mem : ∀ (n : ℕ), g n ∈ 𝒜
hg : ∀ (n : ℕ), (μ (f n ∆ g n)).toReal < ε / (2 * (↑N + 1))
⊢ (μ ((⋃ n ∈ Finset.range (N + 1), f n) ∆ ⋃ n ∈ Finset.range (N + 1), g n)).toReal < ε / 2