Problem 1. Riemann sums do exist for a Riemann integrable function

f(x) on an interval [a, b] with respect to partitions into subintervals that do

not necessarily have equal length. Moreover, for every € > 0 there exists

δ >0 such that the Riemann integral of f over [a, b] is estimated to within

e by the Riemann sum of any partition whose (possibly varying) lengths of

subintervals are all < δ. Work through the following example.