theorem extracted_formal_statement_1.{u_2} {ι : Type u_2} [inst : Nonempty ι] {g h : ι → ℝ} (hg : BddAbove (Set.range g))
  (hh : BddAbove (Set.range h)) (i : ι) : g i + h i ≤ ⨆ j, g i + h j := sorry

State before:
case H
ι : Type u_2
inst✝ : Nonempty ι
g h : ι → ℝ
hg : BddAbove (Set.range g)
hh : BddAbove (Set.range h)
i : ι
⊢ g i + h i ≤ ⨆ j, g i + h j

Tactic: rw [← add_ciSup hh]
    simp only [add_le_add_iff_left]
    exact le_ciSup hh i

State after:

Relevant lemmas:
le_ciSup : ∀ {α : Type u_1} {ι : Sort u_4} [inst : ConditionallyCompleteLattice α] {f : ι → α},
  BddAbove (Set.range f) → ∀ (c : ι), f c ≤ iSup f
add_ciSup : ∀ {ι : Type u_1} {G : Type u_2} [inst : AddGroup G] [inst_1 : ConditionallyCompleteLattice G]
  [inst_2 : Nonempty ι] {f : ι → G} [inst_3 : AddLeftMono G],
  BddAbove (Set.range f) → ∀ (a : G), a + ⨆ i, f i = ⨆ i, a + f i
add_le_add_iff_left : ∀ {α : Type u_1} [inst : Add α] [inst_1 : LE α] [inst_2 : AddLeftMono α]
  [inst_3 : AddLeftReflectLE α] (a : α) {b c : α}, a + b ≤ a + c ↔ b ≤ c