Using Exercise 5.1.12 and the idea of the proof in Exercise 5.1.7, show that Darboux integral is the same as the standard definition of Riemann integral, which you have most likely seen in calculus. That is, show that f∈R([a,b]) if and only if there exists an I∈R, such that for every ϵ>0 there exists a δ>0 such that if P={x0,x1,…,xn} is a partition with Δxi<δ for all i, then ∑i=1nf(ci)Δxi−I<ϵ for every set {c1,c2,…,cn} with ci∈[xi−1,xi]. If f∈R([a,b]), then I=∫abf.