Understanding Rudin Theorem 1.20 (b): Integers and Real Numbers Explained

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In summary, the conversation discusses the proof of the statement that if -m2 < nx < m1 for positive integers m1, m2, n, and real number x, then there exists an integer m between -m2 and m1 that satisfies m-1 ≤ nx < m. The speaker provides a simple demonstration for this statement.
  • #1
Dschumanji
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I understand the proof except for the following:

Suppose that -m2 < nx < m1 for positive integers m1, m2, n, and real number x.

Then there is an integer m with -m2 ≤ m ≤ m1 such that m-1 ≤ nx < m.

It definitely sounds reasonable, but it seems like a big jump in logic.
 
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  • #2
Let m be the least integer that is strictly greater than nx. It is a triviality to verify that this integer has the desired properties.
 
  • #3
Simple proof

Hi there,
I have attached a simple demonstration of the bit you are asking.
Let me know if it is clear now.
I hope it helps
 

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1. What is Rudin Theorem 1.20 (b)?

Rudin Theorem 1.20 (b) is a theorem in the field of mathematical analysis, specifically in the area of real and complex analysis. It states that if a sequence of real or complex numbers converges, then the absolute value of the sequence also converges to the absolute value of the limit.

2. Why is Rudin Theorem 1.20 (b) important?

Rudin Theorem 1.20 (b) is important because it provides a necessary and sufficient condition for the convergence of a sequence. This is a fundamental concept in mathematical analysis and is used in various applications, such as in the study of limits, continuity, and convergence of series.

3. How is Rudin Theorem 1.20 (b) proven?

Rudin Theorem 1.20 (b) can be proven using the definition of convergence and the properties of absolute values. By definition, if a sequence converges, then for any epsilon greater than 0, there exists a natural number N such that the absolute value of the difference between the sequence and its limit is less than epsilon for all n greater than or equal to N. Using this property and the properties of absolute values, we can show that the absolute value of the sequence also converges to the absolute value of the limit.

4. Can Rudin Theorem 1.20 (b) be applied to both real and complex sequences?

Yes, Rudin Theorem 1.20 (b) can be applied to both real and complex sequences. This is because the definition of convergence and the properties of absolute values hold for both real and complex numbers.

5. Are there any limitations to Rudin Theorem 1.20 (b)?

One limitation of Rudin Theorem 1.20 (b) is that it only applies to sequences that converge. It does not provide any information about sequences that do not converge. Additionally, it only applies to sequences of real or complex numbers, and cannot be extended to other types of sequences, such as sequences of functions or matrices.

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