A question from Real and Complex Analysis (Rudin's).

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  • #1
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I am trying to understand theorem 1.17 in page 15-16 international edition 1987.
How do you show that [tex]\phi_n(t)[/tex] is a monotonic increasing sequence of functions?
 
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  • #2
It might be easier if you note that

[tex]k_n(t)=\text{floor}(2^nt)[/tex]

and [tex]\text{floor}(2x)/2\ge \text{floor}(x)[/tex].

Then, when you want to show that [tex]\varphi_n(t)\le\varphi_{n+1}(t)[/tex] consider the cases where [tex]0\le t<n[/tex], [tex]n\le t<n+1[/tex] and [tex]n+1\le t[/tex] separately.
 
  • #3
Thanks, got it, basically I only need to check for t in [0,n) the other case is trivial.
 

1. What is Real and Complex Analysis?

Real and Complex Analysis is a branch of mathematics that deals with the study of real and complex numbers, functions, and their properties. It is a fundamental area of mathematics that has applications in many fields such as physics, engineering, and economics.

2. Who is Rudin and what is his contribution to Real and Complex Analysis?

Walter Rudin was an American mathematician who made significant contributions to the field of Real and Complex Analysis. He is best known for his influential textbooks "Principles of Mathematical Analysis" and "Real and Complex Analysis", which are widely used by students and researchers in the field.

3. What topics are covered in Rudin's "Real and Complex Analysis"?

Rudin's "Real and Complex Analysis" covers a wide range of topics including real and complex numbers, limits, continuity, differentiation, integration, sequences, series, and metric spaces. It also includes more advanced topics such as Lebesgue measure and integration, Fourier analysis, and the theory of analytic functions.

4. Is Rudin's "Real and Complex Analysis" suitable for self-study?

Rudin's "Real and Complex Analysis" is a rigorous and challenging textbook that is often used in advanced undergraduate and graduate courses. It is possible to use it for self-study, but it is recommended to have a strong foundation in calculus and basic analysis before attempting to study from this book.

5. What are some recommended resources for learning Real and Complex Analysis?

Apart from Rudin's "Real and Complex Analysis", there are many other textbooks and online resources available for learning Real and Complex Analysis. Some recommended textbooks include "Real Analysis" by Royden and Fitzpatrick, "Complex Analysis" by Stein and Shakarchi, and "Introduction to Real Analysis" by Bartle and Sherbert. Online resources such as lecture notes, video lectures, and practice problems can also be helpful for supplementing your learning.

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