I am using the 2nd edition of the book page 218 in Calculus by Michael Spivak in the chapter of the Riemann Integral.

$$\sup \{ L(f,P) \} \leq \inf \{ U(f,P) \} $$

It is clear that both of thes numbers are between

$$L(f,P') \leq \sup \{ L(f,P) \} \leq U(f,P')$$

$$L(f,P') \leq \inf \{ U(f,P)\} \leq U(f,P'),$$

for all partitions $P'$

I really do not understand why $L(f,P') \leq \inf \{ U(f,P)\} \leq U(f,P'),$ is true. How does he justify the fact that $U(f,P) \leq U(f,P')$? He didn't state $P' \subset P$, he is claiming that this inequality is true for any partitions $P$ and $P'$. In the previous page he uses $P_1, P_2,P$ so I do not understand where $P'$ suddenly comes from

  • $\begingroup$ Writing $U$ and $L$ as $\sum_{i=1}^{n}M_i\Delta x_i$ and $\sum_{i=1}^{n}m_i\Delta x_i$ and doing comparisions will help. Note that $M_i\ge m_i \forall i$. For comparision over different partitions, P and p', choose only those terms in the definition of L and U that make difference. It is self evident. $\endgroup$ – user45099 Apr 3 '13 at 18:12
  • $\begingroup$ The $P$ in the $\sup,\inf$ is a 'dummy' variable, he uses $P'$ to distinguish the partition from the partitions 'inside' the $\sup, \inf$. It might help to write $\sup_P L(f,P)$ instead. $\endgroup$ – copper.hat Apr 3 '13 at 18:24

Do you understand why $L(f,P') \leq \sup \{ L(f,P) \} \leq U(f,P')$ is true?
Is the same reasoning for $L(f,P') \leq \inf \{ U(f,P)\} \leq U(f,P')$.
The first inequality $L(f,P') \leq \inf \{ U(f,P)\}$ is true because $L(f,P') \leq U(f,P)$ for all partitions $P$ ( If $Q$ is the common refinement of $P$ and $P'$ then $L(f,P') \leq L(f,Q)\leq U(f,Q)\leq U(f,P)$).
For the other inequality note that the infimum is taken over all partitions of your interval (let's denote it as $[a,b]$) therefore $\inf \{ U(f,P)\}=\inf \{ U(f,P):P \text{ partition of }[a,b]\}$ and this is less or equal of $U(f,P')$ since $P'$ is a partition of $[a,b]$.

  • $\begingroup$ No I understand $\sup \{L(f,P) \}$ because $\sup \{L(f,P) \} \leq \inf \{ U(f, P') \} \leq U(f,P') $. $\endgroup$ – Hawk Apr 3 '13 at 18:26
  • $\begingroup$ @sizz: So you don't understand the inequality $\inf\{U(f,P)\}\leq U(f,Q)$. Imagine that you have a nonempty bounded subset $A$ of the reals and $a'\in A$. Then $\inf\{a\in A\}\leq a'$, because $a'\in A$ and $\inf\{a\in A\}$ is a lower bound of $A$, right? Now for the inequality $\inf\{U(f,P)\}\leq U(f,Q)$ the subset $A$ is $A=\{U(f,P):P\text{ partition of }[a,b]\}$ and $a'=U(f,P')$. $\endgroup$ – P.. Apr 3 '13 at 19:03
  • $\begingroup$ Sorry but why do you have $a \in A$ and $a' \in A$? Why don't you just keep one of them? $\endgroup$ – Hawk Apr 3 '13 at 19:11
  • $\begingroup$ What do you mean? For example if $A=\{\frac1n :n\in\mathbb N\}$ and $a'$ is an element of $A$ then $\inf\{a\in A\}\leq a'$. Note that here $\inf\{a\in A\}=\inf\{\frac1n: n\in\mathbb N\}=0$ and $0\leq a'$ for any $a'\in A$. For example for $a'=\frac12$ then $\inf\{a\in A\}\leq \frac12$. $\endgroup$ – P.. Apr 3 '13 at 19:15
  • 1
    $\begingroup$ @sizz: Consider this: If $A=\{1,2,3,4,5,7\}$ then 2 is an element of $A$. Now $\{a\in A\}$ means the set containing all elements in $A$ therefore $\{a\in A\}=\{1,2,3,4,5,7\}$. In this case $\inf\{a\in A\}=\inf\{1,2,3,4,5,7\}=1$ and $1<2$ (2 is an element of $A$ and 1 is the infimum of $A$). $\endgroup$ – P.. Apr 6 '13 at 6:31

For any partition $P$ you have $L(f,P) \le U(f,P)$. If $Q$ is a refinement of $P$, then you have $L(f,P) \le L(f,Q) \le U(f,Q) \le U(f,P)$.

It follows that if $P_1,P_2$ are two partitions, that $L(f,P_1) \le U(f,P_2)$. This is because you can always find a $Q$ that is a refinement of both $P_1,P_2$.

All of the above inequalities follow from this.

For example, since $L(f,P_1) \le U(f,P_2)$, then $L(f,P_1) \le \sup_P L(f,P) \le U(f,P_2)$.

  • $\begingroup$ I do not see how that helps to deduce $U(f,P_1) \leq U(f, P_2)$ $\endgroup$ – Hawk Apr 3 '13 at 18:34
  • $\begingroup$ Because it is not true in general. What is true is that $\inf_P U(f,P) \le U(f,P')$ for any partition $P'$, because the $\inf$ is taken over all partitions (which includes refinements of $P'$). $\endgroup$ – copper.hat Apr 3 '13 at 18:35
  • $\begingroup$ Sorry I am not very familiar with the term "refinement". In Spivak's book (if our notations coincide), I assume you just mean $P_1 \cup P_2 \subset Q$ $\endgroup$ – Hawk Apr 3 '13 at 18:47
  • $\begingroup$ Is it an adequate argument to say that $\inf U(f,P)$ is unique? Therefore since $P'$ and $P$ are partitions of an interval $[a,b]$ we must have either $\inf U(f, P) \leq U(f,P')$ or $\inf U(f,P') \leq U(f,P)$? $\endgroup$ – Hawk Apr 3 '13 at 18:54
  • $\begingroup$ Yes, $A$ refines $B$ if $B \subset A$. I'm not sure what you mean by the $\inf$ is unique. It is, but the fact that you are asking suggests some misunderstanding. I think you are letting the $P$ inside the $\inf$ confuse you. Let $\alpha = \inf_P U(f,P)$. Then you have that $\alpha \le U(f,P')$ for any partition $P'$. $\endgroup$ – copper.hat Apr 3 '13 at 19:00

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