For a nonnegative function defined on an interval define the set
(i.e., the region enclosed by the graph of the function and the -axis between the vertical lines at and ). Then prove
where is the greatest integer less than or equal to .
We can help ourselves by drawing a picture to get a good idea of what is going on, then turn that intuition into something more rigorous. The picture is as follows:
In the picture, we can see the number of lattice points in the ordinate set of (not including the -axis since the question stipulates contains the points ). At each integer between and , we count the number of lattice points beneath , the greatest integer less than or equal to . Then we need to turn this intuition from the picture into a proof:
Proof. Let with . We know such an exists since with . Then, the number of lattice points in with first element is the number of integers such that . But, by definition, this is (the greatest integer less than or equal to ). Summing over all integers we have,