-- Lean 4 proof step from Mathlib.Topology.Order.LiminfLimsup (goal)
Goal:
ι : Type u_1
α : Type u_7
β : Type u_8
inst : ConditionallyCompleteLattice α
inst_1 : CompleteLinearOrder β
inst_2 : TopologicalSpace β
inst_3 : OrderTopology β
u : ι → α → β
c : β
h_all : ∀ (i : ι), Tendsto (u i) atTop (𝓝 c)
h_limsup : Tendsto (fun (r : α) ↦ limsup (fun (i : ι) ↦ u i r) cofinite) atTop (𝓝 c)
h_anti : ∀ (i : ι), Antitone (u i)
⊢ Tendsto (fun (r : α) ↦ ⨆ (i : ι), u i r) atTop (𝓝 c)
have inl : ∀ (hι : IsEmpty ι), Tendsto (fun (r : α) ↦ ⨆ (i : ι), u i r) atTop (𝓝 c)
Tactic: classical
  rcases isEmpty_or_nonempty ι with hι | ⟨⟨n0⟩⟩
  · simpa using h_limsup
  refine tendsto_order.mpr ⟨fun b hb ↦ ?_, fun b hb ↦ ?_⟩
  · filter_upwards with r
    have : c ≤ u n0 r := (h_anti n0).le_of_tendsto (h_all n0) r
    exact hb.trans_le (this.trans (le_iSup_iff.mpr fun b a ↦ a n0))
  -- `⊢ ∀ᶠ (b_1 : α) in atTop, ⨆ i, u i b_1 < b` for `b > c`
  let b' := if h : (Set.Ioo c b).Nonempty then h.some else c
  have hb'b : b' < b := by
    simp only [b']
    split_ifs with h
    exacts [h.some_mem.2, hb]
  have : ∀ᶠ r in atTop, limsup (u · r) cofinite ≤ b' := by
    simp only [b']
    split_ifs with h
    · filter_upwards [(tendsto_order.1 h_limsup).2 _ h.some_mem.1] with r hr using hr.le
    · filter_upwards [(tendsto_order.1 h_limsup).2 b hb] with r hr
      contrapose! h
      exact ⟨limsup (u · r) cofinite, h, hr⟩
  obtain ⟨r, hr⟩ : ∃ r, ∀ s ≥ r, limsup (u · s) cofinite ≤ b' := by simpa using this
  obtain ⟨b'', hb''b, hb''⟩ : ∃ b'' ∈ Set.Ico b' b, ∀ᶠ n in cofinite, u n r ≤ b'' := by
    rcases Set.eq_empty_or_nonempty (Set.Ioo b' b) with h | ⟨b'', hb'b'', hb''b⟩
    · refine ⟨b', ⟨le_rfl, hb'b⟩, ?_⟩
      have h_lt := eventually_lt_of_limsup_lt ((hr r le_rfl).trans_lt hb'b)
      filter_upwards [h_lt] with n hn
      contrapose! h
      exact ⟨u n r, h, hn⟩
    · refine ⟨b'', ⟨hb'b''.le, hb''b⟩ , ?_⟩
      have h_lt := eventually_lt_of_limsup_lt ((hr r le_rfl).trans_lt hb'b'')
      filter_upwards [h_lt] with n hn using hn.le
  have A (n) : ∃ r, ∀ s ≥ r, u n s ≤ b'' := by
    suffices ∀ᶠ r in atTop, u n r ≤ b' by
      simp only [eventually_atTop, ge_iff_le] at this
      rcases this with ⟨r, hr⟩
      exact ⟨r, fun s hs ↦ (hr s hs).trans hb''b.1⟩
    simp only [b']
    split_ifs with h
    · filter_upwards [(tendsto_order.1 (h_all n)).2 _ h.some_mem.1] with r hr
      exact hr.le
    · filter_upwards [(tendsto_order.1 (h_all n)).2 b hb] with r hr
      contrapose! h
      exact ⟨u n r, h, hr⟩
  choose rs hrs using A
  simp only [eventually_atTop, ge_iff_le]
  refine ⟨r ⊔ ⨆ n : {n | b'' < u n r}, rs n, fun v hv ↦ ?_⟩
  -- `⊢ ⨆ i, u i v < b`
  apply lt_of_le_of_lt (iSup_le fun n ↦ ?_) hb''b.2
  -- `⊢ u n v ≤ b''` for `v` such that `r ⊔ (⨆ n, rs n) ≤ v`
  by_cases hn : b'' < u n r
  · refine hrs n v ?_
    calc rs n
    _ = rs (⟨n, by simp [hn]⟩ : {n | b'' < u n r}) := rfl
    _ ≤ ⨆ n : {n | b'' < u n r}, rs n := by
      refine le_ciSup (f := fun (x : {n | b'' < u n r}) ↦ rs x) ?_
        (⟨n, by simp [hn]⟩ : {n | b'' < u n r})
      have : Finite {n | b'' < u n r} := by simpa using hb''
      exact Finite.bddAbove_range _
    _ ≤ r ⊔ ⨆ n : {n | b'' < u n r}, rs n := le_sup_right
    _ ≤ v := hv
  · refine (h_anti n ?_).trans (not_lt.mp hn)
    calc r
    _ ≤ r ⊔ ⨆ n : {n | b'' < u n r}, rs n := le_sup_left
    _ ≤ v := hv