Suppose the inequality presented is true, then so must be the inequality , or alternatively . It can be verified that both these functions are monotonically increasing, positive, and continuous on the specified interval.
Now here is where the geometrically logical but probably incorrect "lemma" I am using comes into play. It goes someting to the tune that if and are monotonic as well as continuous on the interval , then . Now suppose that this is true, it can be easily verified that
Now supposing that my "made up" lemma is correct, this proves the inequality.
I have learned to not always trust my geometric intuition...so Im not too sure about this
EDIT: Wait, they cannot intersect except possibly at the interval because then after that point.
For example but it is reversed afterwards...but you applied my lemma to the interval , and
EDIT EDIT: We dont even need we must just have
EDIT EDIT EDIT: I am busy now but I will come back later and write out this lemma in a clear manner...I will then attempt to prove it
Let for .
Then, we see that is periodic, hence it suffices to prove on .
On the other hand, we have for all , which indicates that it suffices to prove for all .
Clearly, is positive (and decreasing) on , for all , and is negative (and increasing) on , .
Therefore, on .
To complete the proof we have to prove on .
Similar reasoning to the discussion above about increasing and decreasing natures of the functions and together with the fact for all , we get for all , and the proof is hence completed.
.................Graph of .....................................Graph of .....................................Graph of
So the statement is this: Suppose that posses the following charcteristics on : they are positive, they are continuous, they do not intersect, and they are monotonic. Then on it is true that
First let us prove that . Consider any partition of consisting of the set of points . Now as in the usual way define , , and . Now it is clear that since that and since this implies that . Finally we can conclude that . And since are continuous, thus Riemann integrable,
Now let us prove that . Define as before.
1. Now it is clear that either or for all . To see this first define it is clear that is continuous. Then suppose that there were two values such that and then there exists a such that and there exists a such that . Now because is continuous and connected this implies there exists a such that which contradicts that the functions do not intersect.
2. So from the fact that we can see that
3. So all that is left to do is prove that . To do this once again define . Let be the point such that , and let be defined similarly. Now since is compact it follows that . Now consider when are monotonically increasing, it is clear now that . So . So now suppose there was a point such that , then at that point and by the connectedness of and the continuity of there must be a point in such that , but this contradicts the two functions not intersecting. The proof is done similarly for being monotonically decreasing.
4. Now since the interval was arbitrary in 3. this completes the proof
"not intersecting" are only conditions we need: let suppose first that on the interval. it's not hard to see that the integral of a positive continuous function is positive.*
thus conversely, suppose since do not intersect, we have everywhere on [a,b]. so by the intermediate value theorem, either or everywhere
on [a,b]. but if then and hence by * we'll have and hence which is a contradiction. Q.E.D.
* in general, if is continuous, non-negative and not identically 0 on [a,b], then Hint: since is not identically 0 over [a,b], there exists a subinterval of [a,b] over which:
And I understand your proof, but the reason it is so short is that a lot of the stuff you just stated I proved...now of course for a mathematician such as yourself this is obvious...but I thought for us other folks it would be best to show it.
Thanks for your time NonCommAlg