Why infinimum of upper sum and supremum of lower sum will be the only point in between?

Question

Let’s say we have a function $f$, and let $L(f,P)$ denotes the lower sum of $f$ on any partition $P$ and $U(f,P)$ the upper sum of $f$ on $P$. My book writes : If $$sup\{ L(f,P): \text{P is a any partition}\} = inf \{U(f,P) : \text{P is any partition}\}$$ Then they are the only number in between upper and lower sums of $f$ on any partition.

I really don’t know why they will be the only number in between. Can someone help me here?

Let’s say we have a set of partitions $$A = \{P_1, P_2, P_3 .... P_n\}$$ such that $P_2$ have more numbers of points than $P_1$, $P_3$ have more number of points than $P_2$ and so on. $P_n$ contains the largest number of points. Then by the Lemma we know $$ L(f, P_1)\leq L(f,P_2)\leq .... \leq L(f,P_n)$$ and $$ U(f,P_n) \leq U(f, P_{n-1} \leq ... U(f,P_1)$$ if $$L(f,P_n) = U(f, P_n)$$ then also we have so many points/numbers between any two upper and lower sum (of same partition). For example, let’s choose the partition $P_5$, for it we have $$ L(f,P_5) \leq L(f,P_6) ... \leq L(f,P_n) = U(f,P_n) \leq U(f,P_{n-1}) ... \leq U(f,P_5)$$ we got so many numbers between the upper and lower sum of $f$ at partition $P_5$.

Please explain what he meant.

Answer 1

We have $$L(f,P_1) \leq L(f,P_2) ... \leq \sup L(f,P) \leq \inf U(f,P) \leq... U(f,P_2) \leq U(f,P_1),$$ no matter how the partitions are chosen. Both $\sup L(f,P)$ and $\inf L(f,P)$ are greater than or equal to all the lower sums, and both are less than or equal to all the upper sums. If it happens that $\sup L(f,P)=\inf L(f,P)$, then their common value is the only number with this property. (Of course if they're not equal, then any number in between them also has the property.)

I hope this helps.

Answer 2

This has nothing to do with upper or lower sums or partitions.

Obvious result: If a non-empty set $A$ of real numbers is bounded above and another non-empty set $B$ of real numbers is bounded below with $\sup A=\inf B=c\text{ (say)} $ then $c$ is the unique number which lies between elements of $A$ and elements of $B$.

Well first of all there can't be two distinct such elements like $c, d$ with property mentioned above. If there were such numbers with $c<d$ then we have $$a\leq c<d\leq b$$ for all $a\in A, b\in B$. But this implies $\sup A\leq c<d\leq \inf B$ which is contrary to our hypotheses.

On the other hand $c$ possesses this property by definition of supremum and infimum.