Subsequences and Limits in R and R^n .... .... L&S Theorem 5.2 .... ....

In summary, Theorem 5.2 from the book "Real Analysis: Foundations and Functions of One Variable" by Miklos Laczkovich and Vera T. Sos does not have a direct equivalent theorem in \mathbb{R}^n. However, there are related theorems such as the Bolzano-Weierstrass theorem and the Heine-Borel theorem that deal with infinite sequences in higher dimensions. These theorems can be seen as extensions or generalizations of Theorem 5.2.
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In the book " Real Analysis: Foundations and Functions of One Variable" by Miklos Laczkovich and Vera T. Sos, Theorem 5.2 (Chapter 5: Infinite Sequences II) reads as follows:https://www.physicsforums.com/attachments/7722

Can someone inform me if there is an equivalent theorem that holds in \(\displaystyle \mathbb{R}^n\)?Peter
 
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Yes, Theorem 5.2 holds in $\Bbb R^k$, not just $\Bbb R$.
 
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Parker:

As far as I know, there isn't a direct equivalent theorem in \mathbb{R}^n for Theorem 5.2. However, there are some related theorems that deal with infinite sequences in higher dimensions. For example, the Bolzano-Weierstrass theorem states that every bounded sequence in \mathbb{R}^n has a convergent subsequence. This is similar to Theorem 5.2 which deals with bounded sequences in \mathbb{R}. Additionally, there is the Heine-Borel theorem which states that a subset of \mathbb{R}^n is compact if and only if it is closed and bounded. This can be thought of as a higher-dimensional version of the completeness property in Theorem 5.2. Hope this helps!
 

FAQ: Subsequences and Limits in R and R^n .... .... L&S Theorem 5.2 .... ....

1. What is a subsequence in mathematics?

A subsequence is a sequence that is obtained by removing finitely many terms from a given sequence. It may also contain the same terms as the original sequence, but in a different order.

2. What are limits in mathematics?

Limits are used to describe the behavior of a function or sequence as the input values approach a certain point. It is a fundamental concept in calculus and is used to determine the behavior of functions at points where they are not defined or where they are not continuous.

3. What is the L&S Theorem 5.2?

The L&S Theorem 5.2, also known as the Lebesgue's Dominated Convergence Theorem, states that if a sequence of functions converges pointwise and is dominated by an integrable function, then the limit of the sequence is also integrable and the integral of the limit is equal to the limit of the integrals of the sequence.

4. How is the L&S Theorem 5.2 used in mathematics?

The L&S Theorem 5.2 is used to determine the convergence of a sequence of functions and to evaluate the integrals of the limit of the sequence. It is a powerful tool in mathematical analysis, particularly in the field of measure theory.

5. What is the significance of subsequence and limits in R and R^n?

Subsequences and limits are important concepts in R and R^n because they allow us to understand the behavior of functions and sequences in these spaces. They are fundamental to the study of calculus, real analysis, and other branches of mathematics, and have numerous applications in science and engineering.

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