Hello everyone! While doing some problems I came upon this one. What is disconcerting is that I have found (in general) that the more useful a theorem is the more difficult it is to prove. So I would appreciate if someone would tell me if this looks correct?
a partition of containing points with .
is the set of all partitions of
Now suppose that ( is Riemann integrable) on then
Note: by how we defined it follows that for all partitions
Ok now onto the question
Question: Suppose that is a positive, monotonically decreasing function, prove that
Answer: Part one . Consider the interval with . Define the as being the set of natural numbers in . It is clear that . Now consider the th point in the partition. This point will be (since we defined the partitions as the set of natruals). Then the interval will be the interval , so on any interval since is monotonically decreasing. So . So on any interval we have that . So now letting which in turn implies (since there are infinitely many elements of ) so and since the RHS converges by the hypothesis this concludes the proof.
Part two: . Define and as before, consqequently we have that again. Except this time . So . Now as was stated so this implies . So once again letting which in turn implies so . Now since the LHS may be written as and the RHS converges this concludes the proof.
Now combinging parts one and two gives
Hows that look?