Midterm Examination Solutions for Ma 108b Winter
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چکیده
By Theorem 2.1 the product φ1φ2 is of bounded variation and by Theorem 2.24 all three Riemann–Stieltjes integrals involved exist as the limit of Riemann–Stieltjes sums as |Γ| → 0. In particular we can find a δ1 > 0 such that ∣∣∣∣∫ b a fφ2 dφ1 −RΓ(fφ2, φ1) ∣∣∣∣ < ε , ∣∣∣∣∫ b a fφ1 dφ2 −RΓ(fφ1, φ2) ∣∣∣∣ < ε for all partitions with |Γ| < δ1. Let Γ = {xi}i=0 be any partition of [a, b] with intermediate points ξi ∈ [xi−1, xi], then
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