have hf : ∀ (x : ℕ), f x ∈ A := by
  intro x
  simp only [A, f]
  simp only [Set.mem_setOf_eq, le_add_iff_nonneg_right, zero_le]
case left
f : ℕ → ℕ × ℕ := fun n => (n, n + 1)
inj_f : Function.Injective f
⊢ Set.Infinite A
case left
f : ℕ → ℕ × ℕ := fun n => (n, n + 1)
inj_f : Function.Injective f
hf : ∀ (x : ℕ), f x ∈ A
⊢ Set.Infinite A