Proving Lemma 3.3 of L&S: Further Aspects of the Proof

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In summary, Peter is trying to understand Lemma 3.3 of Laczkovich and Sos's book "Real Analysis: Series, Functions of Several Variables, and Applications". He tries several numerical examples to verify that a cube intersects the closure of a given real number ring. However, he is still unsure of how to visualize this. Someone else should help him formalize and rigorously prove this.
  • #1
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I am reading Chapter 3: Jordan Measure ... of Miklos Laczkovich and Vera T Sos's book "Real Analysis: Series, Functions of Several Variables, and Applications" (Springer) ...

I need help with some further aspects of the proof of Lemma 3.3 ... ... in order to fully understand the proof ...

The statement and proof of Lemma 3.3 of L&S reads as follows:
L&S ... Lemma 3.3 ... ...  PART 1 ... .png

L&S ... Lemma 3.3 ... ...  PART 2 ... .... .png


QUESTION 1

In the above proof by L&S we read the following:

" ... ... Let n be fixed. There are integers \(\displaystyle p_j, q_j \) such that

\(\displaystyle \frac{p_j - 1}{n} \lt a_j \leq \frac{p_j}{n} \text{ and } \frac{q_j - 1}{n} \lt b_j \leq \frac{q_j}{n} \)

... ... ... ... "
I tried several numerical examples ... and the examples indicated the above was true ...

... BUT ... ...How do we prove that there exist integers \(\displaystyle p_j, q_j \) such that

\(\displaystyle \frac{p_j - 1}{n} \lt a_j \leq \frac{p_j}{n} \text{ and } \frac{q_j - 1}{n} \lt b_j \leq \frac{q_j}{n} \)

... ... ... ...

QUESTION 2

In the above proof by L&S we read the following:

" ... ... One can easily see that a cube \(\displaystyle [ \frac{ i_1 - 1 }{n} , \frac{ i_1 }{n} ] \ \times \ldots \ \ldots \times \ [ \frac{ i_p - 1 }{n} , \frac{ i_p }{n} ] \) intersects the closure of R ( i.e. R itself) if \(\displaystyle p_j \leq i_j \leq q_j \) (j = 1, ... ... p) ... ...

I am unsure of how to visualize this ...

... ... ... can someone please explain how to formally and rigorously prove that a cube \(\displaystyle [ \frac{ i_1 - 1 }{n} , \frac{ i_1 }{n} ] \ \times \ldots \ \ldots \times \ [ \frac{ i_p - 1 }{n} , \frac{ i_p }{n} ] \) intersects the closure of R ( i.e. R itself) if \(\displaystyle p_j \leq i_j \leq q_j \) (j = 1, ... ... p) ... ... Help with the above two questions will be much appreciated ...

Peter
NOTE:

To make sense of Lemma 3.3 readers of the above post will need access to pages 95-97 of L&S so I providing this text as follows:
L&S ... Jordan Measure ... Ch. 3 ... Page 95 .png

L&S ... Jordan Measure ... Ch. 3 ... Page 96 .... ... .png

L&S ... Jordan Measure ... Ch. 3 ... Page 97 .... ... .png

Hope that helps,

Peter
 
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  • #2
Hi Peter

Question 1

I will prove the existence of $p_{j}$; the argument for $q_{j}$ is similar.

Since the integers are not bounded above, the set $\{p\in\mathbb{Z}\, : \, na_{j}\leq p\}$ is non-empty. By the well-ordering principle, $\{p\in\mathbb{Z}\, : \, na_{j}\leq p\}$ has a least element. Let $p_{j}$ be this least element. Then $p_{j}-1< na_{j}\leq p_{j}\,\Longleftrightarrow\,\dfrac{p_{j}-1}{n}< a_{j}\leq\dfrac{p_{j}}{n}.$

Question 2

Since $p_{j}\leq i_{j}\leq q_{j}\,\Longrightarrow\, na_{j}\leq i_{j}$ and $i_{j}-1\leq nb_{j}$. We must examine two cases.

Case 1: $na_{j}\leq i_{j} \leq nb_{j}$

If this is the case, we are done because $\dfrac{i_{j}}{n}\in \left[\dfrac{i_{j}-1}{n}, \dfrac{i_{j}}{n} \right]\cap [a_{j}, b_{j}]$.

Case 2: $nb_{j} < i_{j}$

In this case, we use $i_{j}-1\leq nb_{j}$ from above. Then $\dfrac{i_{j}-1}{n}\leq b_{j} < \dfrac{i_{j}}{n}\,\Longrightarrow\, b_{j}\in \left[\dfrac{i_{j}-1}{n}, \dfrac{i_{j}}{n} \right]\cap [a_{j}, b_{j}]$.

Feel free to let me know if anything remains unclear.
 
  • #3
Thanks so much for your help GJA …

Still reflecting on what you have written …

… thanks again,

Peter
 

1. How do you prove Lemma 3.3 of L&S: Further Aspects of the Proof?

The proof of Lemma 3.3 involves using mathematical techniques such as induction and logical reasoning. It also requires a thorough understanding of the concepts and definitions presented in L&S: Further Aspects of the Proof.

2. What is the significance of Lemma 3.3 in L&S: Further Aspects of the Proof?

Lemma 3.3 is an important step in the overall proof presented in L&S: Further Aspects of the Proof. It helps establish a key relationship or property that is necessary for the main proof to be valid.

3. Are there any specific assumptions or prerequisites needed for proving Lemma 3.3?

Yes, in order to prove Lemma 3.3, one must have a strong foundation in mathematical concepts and techniques, as well as a thorough understanding of the previous lemmas and theorems presented in L&S: Further Aspects of the Proof.

4. Can Lemma 3.3 be applied to other proofs or areas of study?

Yes, the concepts and techniques used in proving Lemma 3.3 can be applied to other proofs and areas of study in mathematics and related fields. It is a fundamental tool in the world of mathematical proofs.

5. How can one verify the validity of Lemma 3.3?

The validity of Lemma 3.3 can be verified by carefully examining the proof and checking each step for logical consistency and accuracy. It can also be verified by consulting with other experts in the field and comparing it to other established proofs and theorems.

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