Let d be a gauge on I: = (a, b) and suppose that I does not have a d-fine partition.
(a) Let c : = (a + b). Show that at least one of the intervals [a, c] and [c, b] does not have a δ-fine partition.
(b) Construct a nested sequence (In) of subintervals with the length of In equal to (b - a)/2n such that In does not have a δ-fine partition.