But alas, even things that "seem clear" still need formal proof, so ...
Since | P | is a maximum, there must be at least one integer j such that | P | = xj+1 - xj. Take all such points from the partition P, i.e. all points such that | P | = xj+1 - xj. Now consider the refinement P'.
- Suppose none of the additional points are inside the intervals
[xj, xj+1]. Then the original
maximum has not changed so that | P | = | P' |
- Suppose at least one of the additional points is inside at least one of the subintervals [xj, xj+1]. Then this subinterval can no longer contribute to the maximum of P' so that | P' | | P |