proposition: Lemma sum_max : forall b N f, (forall n, f n > 0) -> (sum_f_R0 f b < sum_f_R0 f (max N (S b))).
proof: Proof. intros b N f Hpos. assert (b < max N (S b))%nat by intuition. induction H. replace (sum_f_R0 f b) with (sum_f_R0 f b + 0) by ring. simpl. apply Rplus_le_lt_compat. right. reflexivity. apply Hpos. simpl. replace (sum_f_R0 f b) with (sum_f_R0 f b + 0) by ring. apply Rplus_le_lt_compat. left. apply IHle. apply Hpos. Qed.
imports: PeanoNat Rseries_facts Rsequence_facts Rsequence_subsequence Rpser_usual Reals Rsequence_tactics Rtactic Lra