1. The problem statement, all variables and given/known data Let f [a, b] → R be a non-decreasing function; that is, f(x1) ≤ f(x2) for any x1, x2 ∈ [a, b] with x1 ≤ x2. So f attains a minimum value of m = f(a) and a maximum value of M = f(b) on [a, b]. Let Pn be a regular partition of [a, b] into n subintervals, each of length ∆x = (b − a)/n, and let mi and Mi be the minimum and maximum values of f on the i-th subinterval respectively for each i = 1, 2, . . . , n. (a) Explain why Mi = mi+1 for each i = 1, 2, . . . , n − 1. (b) Hence show that U(f,Pn) − L(f,Pn) = (Mn − m1) ∆x. (c) Express (Mn−m1) ∆x in terms of f, a, b, n and use this to explain why f is integrable on [a, b]. 3. The attempt at a solution A) this is because Mi must be less than mi as it is a non-decreasing function that is why Mi = mi+1 b) =Mi∆x-mi∆x =(Mi-mi)∆x is this part correct? c) (Mn−m1) ∆x (f(a)-f(b))((b-a)/n) how do i show that it is integrable from here?