7.1. Riemann Integral

Examples 7.1.2(c):

Show that if P' is a refinement of P then | P' | | P |
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 | = 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'.

Next | Previous | Glossary | Map