Lb theorem

theorem lb : _lb ≤ SupRegularized fn hl hu := by  -- Suppose, for contradiction, that $\mathrm{InfRegularized} \; fn < \mathrm{SupRegularized} \; fn$.  by_contra h_contra;  -- By definition of `SupRegularized`, we have that $\mathrm{SupRegularized} \; fn \geq \mathrm{InfRegularized} \; fn$.  have h_sup_ge_inf : SupRegularized fn hl hu ≥ InfRegularized fn hl hu := by    apply_rules [ Filter.liminf_le_limsup ];    · exact ⟨ _ub, Filter.eventually_atTop.2 ⟨ 0, fun n hn => hu n ⟩ ⟩;    · exact ⟨ _, Filter.eventually_atTop.2 ⟨ 0, fun n hn => hl n ⟩ ⟩;  exact h_contra ( le_trans ( by exact InfRegularized.lb ) h_sup_ge_inf )