Mono sup

theorem mono_sup (h : Monotone fn) :    SupRegularized fn hl hu = sSup { fn n | n : ℕ} := by  unfold SupRegularized;  simp +decide [ Filter.limsup_eq, Filter.eventually_atTop ];  refine' le_antisymm _ _;  · refine' csInf_le _ _;    · exact ⟨ _lb, by rintro x ⟨ n, hn ⟩ ; exact le_trans ( hl n ) ( hn n le_rfl ) ⟩;    · exact ⟨ 0, fun n _ => le_csSup ⟨ _ub, by rintro x ⟨ m, rfl ⟩ ; exact hu m ⟩ ⟨ n, rfl ⟩ ⟩;  · refine' csSup_le _ _;    · exact ⟨ _, ⟨ 0, rfl ⟩ ⟩;    · norm_num +zetaDelta at *;      exact fun n => le_csInf ⟨ _ub, ⟨ n, fun m hm => hu m ⟩ ⟩ fun x hx => by rcases hx with ⟨ m, hm ⟩ ; exact hm _ ( le_max_left _ _ ) |> le_trans ( h ( le_max_right _ _ ) ) ;