In general, functions such that are not non-increasing. Some of them converge to 0, others don't.
Here's an example. Let me describe the graph of . It consists of triangular peaks that get thinner and higher : around , we put a triangular peak of width and height . (Is it clear?)
The area of the triangle at is . Since , the total area under the curve is finite, i.e. . However, clearly does not converge to 0. We even have for .