proposition: Lemma N_2_epsilon_plus_one:forall eps N, 0 < eps -> 2 / eps< IZR (Z_of_nat N) -> 2/((IZR (Z_of_nat N))+1) <eps.
proof: Proof. intros eps N H_eps H_N. apply Rlt_zero_Rminus. assert (H_N_1:0<(IZR (Z_of_nat N)+1)); [ rewrite_all <- INR_IZR_INZ; rewrite <- S_INR; auto|]. generalize (Rlt_Rminus_zero _ _ H_N); repeat rewrite Rminus_Rdiv_Rmult; auto; clear H_N; intro H_N; unfold Rdiv; apply Rlt_mult_inv_pos; auto. generalize (Rlt_pos_pos_Rdiv _ _ H_eps H_N); clear H_N; intro H_N; apply Rlt_trans with (eps*(IZR (Z_of_nat N))-2); auto; apply Rlt_zero_Rminus; stepr eps; auto; ring. Qed.
imports: Reals QArithSternBrocot(R_addenda) Fourier