Consider a partition of the interval for some .
- Find step functions that are constant on the open subintervals of and integrate to derive the inequalities:
- Give a geometric interpretation of the inequalities in part (a).
- Find a particular partition (i.e., choose particular values for ) to establish the following inequalities for ,
- Proof. We define step function and by
Since is strictly decreasing on , we have
Therefore, using the definition of the integral of a step function as a sum,
- Geometrically, these inequalities say that the area under the curve lies between the step functions that take on the values and for each .
- Proof. To establish these inequalities we pick the partition,
Then, applying part (a) we have
The final line follows since so the sum on the left starts with and the sum on the right only runs to . These were the inequalities requested
In part c, if you define P = {1,2,3, …, n} then only a_0 through a_(n-1) are defined, the left sum refers to a_n which is not defined in P.