-- Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean
lemma measurableSet_bddAbove_range {ι} [Countable ι] {f : ι → δ → α} (hf : ∀ i, Measurable (f i)) :
    MeasurableSet {b | BddAbove (range (fun i ↦ f i b))}

Proof state:
case inr
α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), Measurable (f i)
hα : Nonempty α
A : ∀ (i : ι) (c : α), MeasurableSet {x | f i x ≤ c}
B : ∀ (c : α), MeasurableSet {x | ∀ (i : ι), f i x ≤ c}
⊢ MeasurableSet {b | BddAbove (range fun i => f i b)}

Tactic:
obtain ⟨u, hu⟩ : ∃ (u : ℕ → α), Tendsto u atTop atTop := exists_seq_tendsto (atTop : Filter α)

Result:
case inr.intro
α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), Measurable (f i)
hα : Nonempty α
A : ∀ (i : ι) (c : α), MeasurableSet {x | f i x ≤ c}
B : ∀ (c : α), MeasurableSet {x | ∀ (i : ι), f i x ≤ c}
u : ℕ → α
hu : Tendsto u atTop atTop
⊢ MeasurableSet {b | BddAbove (range fun i => f i b)}

Proof state:
case inr
α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), Measurable (f i)
hα : Nonempty α
A : ∀ (i : ι) (c : α), MeasurableSet {x | f i x ≤ c}
⊢ MeasurableSet {b | BddAbove (range fun i => f i b)}

Tactic:
have B : ∀ (c : α), MeasurableSet {x | ∀ i, f i x ≤ c} := by
  intro c
  rw [setOf_forall]
  exact MeasurableSet.iInter (fun i ↦ A i c)

Result:
case inr
α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), Measurable (f i)
hα : Nonempty α
A : ∀ (i : ι) (c : α), MeasurableSet {x | f i x ≤ c}
B : ∀ (c : α), MeasurableSet {x | ∀ (i : ι), f i x ≤ c}
⊢ MeasurableSet {b | BddAbove (range fun i => f i b)}

Proof state:
α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), Measurable (f i)
hα : Nonempty α
⊢ ∀ (i : ι) (c : α), MeasurableSet {x | f i x ≤ c}

Tactic:
intro i c

Result:
α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), Measurable (f i)
hα : Nonempty α
i : ι
c : α
⊢ MeasurableSet {x | f i x ≤ c}

Proof state:
case h₁
α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), Measurable (f i)
hα : Nonempty α
A : ∀ (i : ι) (c : α), MeasurableSet {x | f i x ≤ c}
B : ∀ (c : α), MeasurableSet {x | ∀ (i : ι), f i x ≤ c}
u : ℕ → α
hu : Tendsto u atTop atTop
⊢ {b | BddAbove (range fun i => f i b)} ⊆ {x | ∃ n, ∀ (i : ι), f i x ≤ u n}

Tactic:
rintro x ⟨c, hc⟩

Result:
case h₁.intro
α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), Measurable (f i)
hα : Nonempty α
A : ∀ (i : ι) (c : α), MeasurableSet {x | f i x ≤ c}
B : ∀ (c : α), MeasurableSet {x | ∀ (i : ι), f i x ≤ c}
u : ℕ → α
hu : Tendsto u atTop atTop
x : δ
c : α
hc : c ∈ upperBounds (range fun i => f i x)
⊢ x ∈ {x | ∃ n, ∀ (i : ι), f i x ≤ u n}

Proof state:
case inr
α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), Measurable (f i)
hα : Nonempty α
⊢ MeasurableSet {b | BddAbove (range fun i => f i b)}

Tactic:
have A : ∀ (i : ι) (c : α), MeasurableSet {x | f i x ≤ c} := by
  intro i c
  exact measurableSet_le (hf i) measurable_const

Result:
case inr
α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), Measurable (f i)
hα : Nonempty α
A : ∀ (i : ι) (c : α), MeasurableSet {x | f i x ≤ c}
⊢ MeasurableSet {b | BddAbove (range fun i => f i b)}

Proof state:
case h₂
α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), Measurable (f i)
hα : Nonempty α
A : ∀ (i : ι) (c : α), MeasurableSet {x | f i x ≤ c}
B : ∀ (c : α), MeasurableSet {x | ∀ (i : ι), f i x ≤ c}
u : ℕ → α
hu : Tendsto u atTop atTop
⊢ {x | ∃ n, ∀ (i : ι), f i x ≤ u n} ⊆ {b | BddAbove (range fun i => f i b)}

Tactic:
rintro x ⟨n, hn⟩

Result:
case h₂.intro
α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), Measurable (f i)
hα : Nonempty α
A : ∀ (i : ι) (c : α), MeasurableSet {x | f i x ≤ c}
B : ∀ (c : α), MeasurableSet {x | ∀ (i : ι), f i x ≤ c}
u : ℕ → α
hu : Tendsto u atTop atTop
x : δ
n : ℕ
hn : ∀ (i : ι), f i x ≤ u n
⊢ x ∈ {b | BddAbove (range fun i => f i b)}

Proof state:
case inl
α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), Measurable (f i)
hα : IsEmpty α
⊢ MeasurableSet {b | BddAbove (range fun i => f i b)}

Tactic:
have : ∀ b, range (fun i ↦ f i b) = ∅ := fun b ↦ eq_empty_of_isEmpty _

Result:
case inl
α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), Measurable (f i)
hα : IsEmpty α
this : ∀ (b : δ), (range fun i => f i b) = ∅
⊢ MeasurableSet {b | BddAbove (range fun i => f i b)}