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

[Math Amateur]

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

 

[Euge]

Yes, Theorem 5.2 holds in $\Bbb R^k$, not just $\Bbb R$.

 

[math771]

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!