## 7.1. Riemann Integral

### Examples 7.1.2(c):

To prove this fact is more confusing than enlightening. It seems clear that if one or more points are inserted into the partition*P*to form the refinement partition

*P'*, the largest distance between the points of

*P'*must now be less than (or equal to) that of the points of

*P*.

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 | = x _{j+1} - x_{j}*. Take all such
points from the partition

*P*, i.e. all points such that

*| P | = x*. Now consider the refinement

_{j+1}- x_{j}*P'*.

- Suppose none of the additional points are inside the intervals
*[x*. Then the original maximum has not changed so that_{j}, x_{j+1}]*| P | = | P' |* - Suppose at least one of the additional points is inside at
least one of the subintervals
*[x*. Then this subinterval can no longer contribute to the maximum of_{j}, x_{j+1}]*P'*so that*| P' | | P |*