Exercise 11, § 1 Functions of a Real Variable, II, 81 With the same hypotheses as in exerc. 10 show that for every ε>0\varepsilon > 0ε>0 there exists a number ρ>0\rho > 0ρ>0 such that for every subdivision of [a, b][a,\ b][a, b] into intervals [xi, xi+1][x_i,\ x_{i+1}][xi, xi+1] of length ≤ρ\leq \rho≤ρ (0≤i≤n−10 \leq i \leq n-10≤i≤n−1) one has ∣∫ab∣f(t).g(t)∣ dt−∑i=0n−1[f(ui).g(vi)](xi+1−xi)∣≤ε \left| \int_a^b |\mathbf{f}(t).\mathbf{g}(t)|\,dt - \sum_{i=0}^{n-1} [\mathbf{f}(u_i).\mathbf{g}(v_i)] (x_{i+1} - x_i) \right| \leq \varepsilon ∫ab∣f(t).g(t)∣dt−i=0∑n−1[f(ui).g(vi)](xi+1−xi)≤ε for any choice, for each index iii, of points ui, viu_i,\ v_iui, vi in [xi, xi+1][x_i,\ x_{i+1}][xi, xi+1] (reduce to the case where f\mathbf{f}f and g\mathbf{g}g are step functions). Solution No solution has been written yet.