feat(topology/algebra/order/compact): remove conditional completeness…

… assumption in `is_compact.exists_forall_le` (#18991)

src/order/directed.lean

@@ -131,6 +131,10 @@ lemma directed_on_of_inf_mem [semilattice_inf α] {S : set α}
   (H : ∀ ⦃i j⦄, i ∈ S → j ∈ S → i ⊓ j ∈ S) : directed_on (≥) S :=
 λ a ha b hb, ⟨a ⊓ b, H ha hb, inf_le_left, inf_le_right⟩
 
+lemma is_total.directed [is_total α r] (f : ι → α) :
+  directed r f :=
+λ i j, or.cases_on (total_of r (f i) (f j)) (λ h, ⟨j, h, refl _⟩) (λ h, ⟨i, refl _, h⟩)
+
 /-- `is_directed α r` states that for any elements `a`, `b` there exists an element `c` such that
 `r a c` and `r b c`. -/
 class is_directed (α : Type*) (r : α → α → Prop) : Prop :=
@@ -150,7 +154,7 @@ lemma directed_on_univ_iff : directed_on r set.univ ↔ is_directed α r :=
 
 @[priority 100]  -- see Note [lower instance priority]
 instance is_total.to_is_directed [is_total α r] : is_directed α r :=
-⟨λ a b, or.cases_on (total_of r a b) (λ h, ⟨b, h, refl _⟩) (λ h, ⟨a, refl _, h⟩)⟩
+by rw ← directed_id_iff; exact is_total.directed _
 
 lemma is_directed_mono [is_directed α r] (h : ∀ ⦃a b⦄, r a b → s a b) : is_directed α s :=
 ⟨λ a b, let ⟨c, ha, hb⟩ := is_directed.directed a b in ⟨c, h ha, h hb⟩⟩

src/topology/algebra/order/compact.lean

@@ -136,80 +136,39 @@ is_compact_iff_compact_space.mp is_compact_Icc
 end
 
 /-!
-### Min and max elements of a compact set
+### Extreme value theorem
 -/
 
-variables {α β γ : Type*} [conditionally_complete_linear_order α] [topological_space α]
-  [order_topology α] [topological_space β] [topological_space γ]
-
-lemma is_compact.Inf_mem {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
-  Inf s ∈ s :=
-hs.is_closed.cInf_mem ne_s hs.bdd_below
-
-lemma is_compact.Sup_mem {s : set α} (hs : is_compact s) (ne_s : s.nonempty) : Sup s ∈ s :=
-@is_compact.Inf_mem αᵒᵈ _ _ _ _ hs ne_s
-
-lemma is_compact.is_glb_Inf {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
-  is_glb s (Inf s) :=
-is_glb_cInf ne_s hs.bdd_below
-
-lemma is_compact.is_lub_Sup {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
-  is_lub s (Sup s) :=
-@is_compact.is_glb_Inf αᵒᵈ _ _ _ _ hs ne_s
-
-lemma is_compact.is_least_Inf {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
-  is_least s (Inf s) :=
-⟨hs.Inf_mem ne_s, (hs.is_glb_Inf ne_s).1⟩
+section linear_order
 
-lemma is_compact.is_greatest_Sup {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
-  is_greatest s (Sup s) :=
-@is_compact.is_least_Inf αᵒᵈ _ _ _ _ hs ne_s
+variables {α β γ : Type*} [linear_order α] [topological_space α]
+  [order_closed_topology α] [topological_space β] [topological_space γ]
 
 lemma is_compact.exists_is_least {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
   ∃ x, is_least s x :=
-⟨_, hs.is_least_Inf ne_s⟩
+begin
+  haveI : nonempty s := ne_s.to_subtype,
+  suffices : (s ∩ ⋂ x ∈ s, Iic x).nonempty,
+    from ⟨this.some, this.some_spec.1, mem_Inter₂.mp this.some_spec.2⟩,
+  rw bInter_eq_Inter,
+  by_contra H,
+  rw not_nonempty_iff_eq_empty at H,
+  rcases hs.elim_directed_family_closed (λ x : s, Iic ↑x) (λ x, is_closed_Iic) H
+    ((is_total.directed coe).mono_comp _ (λ _ _, Iic_subset_Iic.mpr)) with ⟨x, hx⟩,
+  exact not_nonempty_iff_eq_empty.mpr hx ⟨x, x.2, le_rfl⟩
+end
 
 lemma is_compact.exists_is_greatest {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
   ∃ x, is_greatest s x :=
-⟨_, hs.is_greatest_Sup ne_s⟩
+@is_compact.exists_is_least αᵒᵈ _ _ _ _ hs ne_s
 
 lemma is_compact.exists_is_glb {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
   ∃ x ∈ s, is_glb s x :=
-⟨_, hs.Inf_mem ne_s, hs.is_glb_Inf ne_s⟩
+exists_imp_exists (λ x (hx : is_least s x), ⟨hx.1, hx.is_glb⟩) (hs.exists_is_least ne_s)
 
 lemma is_compact.exists_is_lub {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
   ∃ x ∈ s, is_lub s x :=
-⟨_, hs.Sup_mem ne_s, hs.is_lub_Sup ne_s⟩
-
-lemma is_compact.exists_Inf_image_eq_and_le {s : set β} (hs : is_compact s) (ne_s : s.nonempty)
-  {f : β → α} (hf : continuous_on f s) :
-  ∃ x ∈ s, Inf (f '' s) = f x ∧ ∀ y ∈ s, f x ≤ f y :=
-let ⟨x, hxs, hx⟩ := (hs.image_of_continuous_on hf).Inf_mem (ne_s.image f)
-in ⟨x, hxs, hx.symm, λ y hy,
-  hx.trans_le $ cInf_le (hs.image_of_continuous_on hf).bdd_below $ mem_image_of_mem f hy⟩
-
-lemma is_compact.exists_Sup_image_eq_and_ge {s : set β} (hs : is_compact s) (ne_s : s.nonempty)
-  {f : β → α} (hf : continuous_on f s) :
-  ∃ x ∈ s, Sup (f '' s) = f x ∧ ∀ y ∈ s, f y ≤ f x :=
-@is_compact.exists_Inf_image_eq_and_le αᵒᵈ _ _ _ _ _ _ hs ne_s _ hf
-
-lemma is_compact.exists_Inf_image_eq {s : set β} (hs : is_compact s) (ne_s : s.nonempty)
-  {f : β → α} (hf : continuous_on f s) :
-  ∃ x ∈ s,  Inf (f '' s) = f x :=
-let ⟨x, hxs, hx, _⟩ := hs.exists_Inf_image_eq_and_le ne_s hf in ⟨x, hxs, hx⟩
-
-lemma is_compact.exists_Sup_image_eq :
-  ∀ {s : set β}, is_compact s → s.nonempty → ∀ {f : β → α}, continuous_on f s →
-  ∃ x ∈ s, Sup (f '' s) = f x :=
-@is_compact.exists_Inf_image_eq αᵒᵈ _ _ _ _ _
-
-lemma eq_Icc_of_connected_compact {s : set α} (h₁ : is_connected s) (h₂ : is_compact s) :
-  s = Icc (Inf s) (Sup s) :=
-eq_Icc_cInf_cSup_of_connected_bdd_closed h₁ h₂.bdd_below h₂.bdd_above h₂.is_closed
-
-/-!
-### Extreme value theorem
--/
+@is_compact.exists_is_glb αᵒᵈ _ _ _ _ hs ne_s
 
 /-- The **extreme value theorem**: a continuous function realizes its minimum on a compact set. -/
 lemma is_compact.exists_forall_le {s : set β} (hs : is_compact s) (ne_s : s.nonempty)
@@ -277,6 +236,68 @@ lemma continuous.exists_forall_ge [nonempty β] {f : β → α}
   ∃ x, ∀ y, f y ≤ f x :=
 @continuous.exists_forall_le αᵒᵈ _ _ _ _ _ _ _ hf hlim
 
+/-- A continuous function with compact support has a global minimum. -/
+@[to_additive "A continuous function with compact support has a global minimum."]
+lemma continuous.exists_forall_le_of_has_compact_mul_support [nonempty β] [has_one α]
+  {f : β → α} (hf : continuous f) (h : has_compact_mul_support f) :
+  ∃ (x : β), ∀ (y : β), f x ≤ f y :=
+begin
+  obtain ⟨_, ⟨x, rfl⟩, hx⟩ := (h.is_compact_range hf).exists_is_least (range_nonempty _),
+  rw [mem_lower_bounds, forall_range_iff] at hx,
+  exact ⟨x, hx⟩,
+end
+
+/-- A continuous function with compact support has a global maximum. -/
+@[to_additive "A continuous function with compact support has a global maximum."]
+lemma continuous.exists_forall_ge_of_has_compact_mul_support [nonempty β] [has_one α]
+  {f : β → α} (hf : continuous f) (h : has_compact_mul_support f) :
+  ∃ (x : β), ∀ (y : β), f y ≤ f x :=
+@continuous.exists_forall_le_of_has_compact_mul_support αᵒᵈ _ _ _ _ _ _ _ _ hf h
+
+/-- A compact set is bounded below -/
+lemma is_compact.bdd_below [nonempty α] {s : set α} (hs : is_compact s) : bdd_below s :=
+begin
+  cases s.eq_empty_or_nonempty,
+  { rw h,
+    exact bdd_below_empty },
+  { obtain ⟨a, ha, has⟩ := hs.exists_is_least h,
+    exact ⟨a, has⟩ },
+end
+
+/-- A compact set is bounded above -/
+lemma is_compact.bdd_above [nonempty α] {s : set α} (hs : is_compact s) : bdd_above s :=
+@is_compact.bdd_below αᵒᵈ _ _ _ _ _ hs
+
+/-- A continuous function is bounded below on a compact set. -/
+lemma is_compact.bdd_below_image [nonempty α] {f : β → α} {K : set β}
+  (hK : is_compact K) (hf : continuous_on f K) : bdd_below (f '' K) :=
+(hK.image_of_continuous_on hf).bdd_below
+
+/-- A continuous function is bounded above on a compact set. -/
+lemma is_compact.bdd_above_image [nonempty α] {f : β → α} {K : set β}
+  (hK : is_compact K) (hf : continuous_on f K) : bdd_above (f '' K) :=
+@is_compact.bdd_below_image αᵒᵈ _ _ _ _ _ _ _ _ hK hf
+
+/-- A continuous function with compact support is bounded below. -/
+@[to_additive /-" A continuous function with compact support is bounded below. "-/]
+lemma continuous.bdd_below_range_of_has_compact_mul_support [has_one α] {f : β → α}
+  (hf : continuous f) (h : has_compact_mul_support f) : bdd_below (range f) :=
+(h.is_compact_range hf).bdd_below
+
+/-- A continuous function with compact support is bounded above. -/
+@[to_additive /-" A continuous function with compact support is bounded above. "-/]
+lemma continuous.bdd_above_range_of_has_compact_mul_support [has_one α]
+  {f : β → α} (hf : continuous f) (h : has_compact_mul_support f) :
+  bdd_above (range f) :=
+@continuous.bdd_below_range_of_has_compact_mul_support αᵒᵈ _ _ _ _ _ _ _ hf h
+
+end linear_order
+
+section conditionally_complete_linear_order
+
+variables {α β γ : Type*} [conditionally_complete_linear_order α] [topological_space α]
+  [order_closed_topology α] [topological_space β] [topological_space γ]
+
 lemma is_compact.Sup_lt_iff_of_continuous {f : β → α}
   {K : set β} (hK : is_compact K) (h0K : K.nonempty) (hf : continuous_on f K) (y : α) :
   Sup (f '' K) < y ↔ ∀ x ∈ K, f x < y :=
@@ -294,23 +315,71 @@ lemma is_compact.lt_Inf_iff_of_continuous {α β : Type*}
   y < Inf (f '' K) ↔ ∀ x ∈ K, y < f x :=
 @is_compact.Sup_lt_iff_of_continuous αᵒᵈ β _ _ _ _ _ _ hK h0K hf y
 
-/-- A continuous function with compact support has a global minimum. -/
-@[to_additive "A continuous function with compact support has a global minimum."]
-lemma continuous.exists_forall_le_of_has_compact_mul_support [nonempty β] [has_one α]
-  {f : β → α} (hf : continuous f) (h : has_compact_mul_support f) :
-  ∃ (x : β), ∀ (y : β), f x ≤ f y :=
-begin
-  obtain ⟨_, ⟨x, rfl⟩, hx⟩ := (h.is_compact_range hf).exists_is_least (range_nonempty _),
-  rw [mem_lower_bounds, forall_range_iff] at hx,
-  exact ⟨x, hx⟩,
-end
+end conditionally_complete_linear_order
 
-/-- A continuous function with compact support has a global maximum. -/
-@[to_additive "A continuous function with compact support has a global maximum."]
-lemma continuous.exists_forall_ge_of_has_compact_mul_support [nonempty β] [has_one α]
-  {f : β → α} (hf : continuous f) (h : has_compact_mul_support f) :
-  ∃ (x : β), ∀ (y : β), f y ≤ f x :=
-@continuous.exists_forall_le_of_has_compact_mul_support αᵒᵈ _ _ _ _ _ _ _ _ hf h
+/-!
+### Min and max elements of a compact set
+-/
+
+section order_closed_topology
+
+variables {α β γ : Type*} [conditionally_complete_linear_order α] [topological_space α]
+  [order_closed_topology α] [topological_space β] [topological_space γ]
+
+lemma is_compact.Inf_mem {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
+  Inf s ∈ s :=
+let ⟨a, ha⟩ := hs.exists_is_least ne_s in
+ha.Inf_mem
+
+lemma is_compact.Sup_mem {s : set α} (hs : is_compact s) (ne_s : s.nonempty) : Sup s ∈ s :=
+@is_compact.Inf_mem αᵒᵈ _ _ _ _ hs ne_s
+
+lemma is_compact.is_glb_Inf {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
+  is_glb s (Inf s) :=
+is_glb_cInf ne_s hs.bdd_below
+
+lemma is_compact.is_lub_Sup {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
+  is_lub s (Sup s) :=
+@is_compact.is_glb_Inf αᵒᵈ _ _ _ _ hs ne_s
+
+lemma is_compact.is_least_Inf {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
+  is_least s (Inf s) :=
+⟨hs.Inf_mem ne_s, (hs.is_glb_Inf ne_s).1⟩
+
+lemma is_compact.is_greatest_Sup {s : set α} (hs : is_compact s) (ne_s : s.nonempty) :
+  is_greatest s (Sup s) :=
+@is_compact.is_least_Inf αᵒᵈ _ _ _ _ hs ne_s
+
+lemma is_compact.exists_Inf_image_eq_and_le {s : set β} (hs : is_compact s) (ne_s : s.nonempty)
+  {f : β → α} (hf : continuous_on f s) :
+  ∃ x ∈ s, Inf (f '' s) = f x ∧ ∀ y ∈ s, f x ≤ f y :=
+let ⟨x, hxs, hx⟩ := (hs.image_of_continuous_on hf).Inf_mem (ne_s.image f)
+in ⟨x, hxs, hx.symm, λ y hy,
+  hx.trans_le $ cInf_le (hs.image_of_continuous_on hf).bdd_below $ mem_image_of_mem f hy⟩
+
+lemma is_compact.exists_Sup_image_eq_and_ge {s : set β} (hs : is_compact s) (ne_s : s.nonempty)
+  {f : β → α} (hf : continuous_on f s) :
+  ∃ x ∈ s, Sup (f '' s) = f x ∧ ∀ y ∈ s, f y ≤ f x :=
+@is_compact.exists_Inf_image_eq_and_le αᵒᵈ _ _ _ _ _ _ hs ne_s _ hf
+
+lemma is_compact.exists_Inf_image_eq {s : set β} (hs : is_compact s) (ne_s : s.nonempty)
+  {f : β → α} (hf : continuous_on f s) :
+  ∃ x ∈ s,  Inf (f '' s) = f x :=
+let ⟨x, hxs, hx, _⟩ := hs.exists_Inf_image_eq_and_le ne_s hf in ⟨x, hxs, hx⟩
+
+lemma is_compact.exists_Sup_image_eq :
+  ∀ {s : set β}, is_compact s → s.nonempty → ∀ {f : β → α}, continuous_on f s →
+  ∃ x ∈ s, Sup (f '' s) = f x :=
+@is_compact.exists_Inf_image_eq αᵒᵈ _ _ _ _ _
+
+end order_closed_topology
+
+variables {α β γ : Type*} [conditionally_complete_linear_order α] [topological_space α]
+  [order_topology α] [topological_space β] [topological_space γ]
+
+lemma eq_Icc_of_connected_compact {s : set α} (h₁ : is_connected s) (h₂ : is_compact s) :
+  s = Icc (Inf s) (Sup s) :=
+eq_Icc_cInf_cSup_of_connected_bdd_closed h₁ h₂.bdd_below h₂.bdd_above h₂.is_closed
 
 lemma is_compact.continuous_Sup {f : γ → β → α}
   {K : set β} (hK : is_compact K) (hf : continuous ↿f) :

src/topology/order/basic.lean

@@ -648,48 +648,6 @@ lemma dense.exists_between [densely_ordered α] {s : set α} (hs : dense s) {x y
   ∃ z ∈ s, z ∈ Ioo x y :=
 hs.exists_mem_open is_open_Ioo (nonempty_Ioo.2 h)
 
-variables [nonempty α] [topological_space β]
-
-/-- A compact set is bounded below -/
-lemma is_compact.bdd_below {s : set α} (hs : is_compact s) : bdd_below s :=
-begin
-  by_contra H,
-  rcases hs.elim_finite_subcover_image (λ x (_ : x ∈ s), @is_open_Ioi _ _ _ _ x) _
-    with ⟨t, st, ft, ht⟩,
-  { refine H (ft.bdd_below.imp $ λ C hC y hy, _),
-    rcases mem_Union₂.1 (ht hy) with ⟨x, hx, xy⟩,
-    exact le_trans (hC hx) (le_of_lt xy) },
-  { refine λ x hx, mem_Union₂.2 (not_imp_comm.1 _ H),
-    exact λ h, ⟨x, λ y hy, le_of_not_lt (h.imp $ λ ys, ⟨_, hy, ys⟩)⟩ }
-end
-
-/-- A compact set is bounded above -/
-lemma is_compact.bdd_above {s : set α} (hs : is_compact s) : bdd_above s :=
-@is_compact.bdd_below αᵒᵈ _ _ _ _ _ hs
-
-/-- A continuous function is bounded below on a compact set. -/
-lemma is_compact.bdd_below_image {f : β → α} {K : set β}
-  (hK : is_compact K) (hf : continuous_on f K) : bdd_below (f '' K) :=
-(hK.image_of_continuous_on hf).bdd_below
-
-/-- A continuous function is bounded above on a compact set. -/
-lemma is_compact.bdd_above_image {f : β → α} {K : set β}
-  (hK : is_compact K) (hf : continuous_on f K) : bdd_above (f '' K) :=
-@is_compact.bdd_below_image αᵒᵈ _ _ _ _ _ _ _ _ hK hf
-
-/-- A continuous function with compact support is bounded below. -/
-@[to_additive /-" A continuous function with compact support is bounded below. "-/]
-lemma continuous.bdd_below_range_of_has_compact_mul_support [has_one α] {f : β → α}
-  (hf : continuous f) (h : has_compact_mul_support f) : bdd_below (range f) :=
-(h.is_compact_range hf).bdd_below
-
-/-- A continuous function with compact support is bounded above. -/
-@[to_additive /-" A continuous function with compact support is bounded above. "-/]
-lemma continuous.bdd_above_range_of_has_compact_mul_support [has_one α]
-  {f : β → α} (hf : continuous f) (h : has_compact_mul_support f) :
-  bdd_above (range f) :=
-@continuous.bdd_below_range_of_has_compact_mul_support αᵒᵈ _ _ _ _ _ _ _ _ hf h
-
 end linear_order
 
 end order_closed_topology

src/topology/subset_properties.lean

@@ -230,6 +230,19 @@ lemma is_compact.disjoint_nhds_set_right {l : filter α} (hs : is_compact s) :
   disjoint l (𝓝ˢ s) ↔ ∀ x ∈ s, disjoint l (𝓝 x) :=
 by simpa only [disjoint.comm] using hs.disjoint_nhds_set_left
 
+/-- For every directed family of closed sets whose intersection avoids a compact set,
+there exists a single element of the family which itself avoids this compact set. -/
+lemma is_compact.elim_directed_family_closed {ι : Type v} [hι : nonempty ι] (hs : is_compact s)
+  (Z : ι → set α) (hZc : ∀ i, is_closed (Z i)) (hsZ : s ∩ (⋂ i, Z i) = ∅) (hdZ : directed (⊇) Z) :
+  ∃ i : ι, s ∩ Z i = ∅ :=
+let ⟨t, ht⟩ := hs.elim_directed_cover (compl ∘ Z) (λ i, (hZc i).is_open_compl)
+  (by simpa only [subset_def, not_forall, eq_empty_iff_forall_not_mem, mem_Union,
+    exists_prop, mem_inter_iff, not_and, iff_self, mem_Inter, mem_compl_iff] using hsZ)
+  (hdZ.mono_comp _ $ λ _ _, compl_subset_compl.mpr)
+    in
+⟨t, by simpa only [subset_def, not_forall, eq_empty_iff_forall_not_mem, mem_Union,
+    exists_prop, mem_inter_iff, not_and, iff_self, mem_Inter, mem_compl_iff] using ht⟩
+
 /-- For every family of closed sets whose intersection avoids a compact set,
 there exists a finite subfamily whose intersection avoids this compact set. -/
 lemma is_compact.elim_finite_subfamily_closed {s : set α} {ι : Type v} (hs : is_compact s)
@@ -273,25 +286,16 @@ lemma is_compact.nonempty_Inter_of_directed_nonempty_compact_closed
   (hZn : ∀ i, (Z i).nonempty) (hZc : ∀ i, is_compact (Z i)) (hZcl : ∀ i, is_closed (Z i)) :
   (⋂ i, Z i).nonempty :=
 begin
-  apply hι.elim,
-  intro i₀,
-  let Z' := λ i, Z i ∩ Z i₀,
-  suffices : (⋂ i, Z' i).nonempty,
-  { exact this.mono (Inter_mono $ λ i, inter_subset_left (Z i) (Z i₀)) },
-  rw nonempty_iff_ne_empty,
-  intro H,
-  obtain ⟨t, ht⟩ : ∃ (t : finset ι), ((Z i₀) ∩ ⋂ (i ∈ t), Z' i) = ∅,
-    from (hZc i₀).elim_finite_subfamily_closed Z'
-      (assume i, is_closed.inter (hZcl i) (hZcl i₀)) (by rw [H, inter_empty]),
-  obtain ⟨i₁, hi₁⟩ : ∃ i₁ : ι, Z i₁ ⊆ Z i₀ ∧ ∀ i ∈ t, Z i₁ ⊆ Z' i,
-  { rcases directed.finset_le hZd t with ⟨i, hi⟩,
-    rcases hZd i i₀ with ⟨i₁, hi₁, hi₁₀⟩,
-    use [i₁, hi₁₀],
-    intros j hj,
-    exact subset_inter (subset.trans hi₁ (hi j hj)) hi₁₀ },
-  suffices : ((Z i₀) ∩ ⋂ (i ∈ t), Z' i).nonempty,
-  { rw nonempty_iff_ne_empty at this, contradiction },
-  exact (hZn i₁).mono (subset_inter hi₁.left $ subset_Inter₂ hi₁.right),
+  let i₀ := hι.some,
+  suffices : (Z i₀ ∩ ⋂ i, Z i).nonempty,
+    by rwa inter_eq_right_iff_subset.mpr (Inter_subset _ i₀) at this,
+  simp only [nonempty_iff_ne_empty] at hZn ⊢,
+  apply mt ((hZc i₀).elim_directed_family_closed Z hZcl),
+  push_neg,
+  simp only [← nonempty_iff_ne_empty] at hZn ⊢,
+  refine ⟨hZd, λ i, _⟩,
+  rcases hZd i₀ i with ⟨j, hji₀, hji⟩,
+  exact (hZn j).mono (subset_inter hji₀ hji)
 end
 
 /-- Cantor's intersection theorem for sequences indexed by `ℕ`: