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

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Theorem 5.2 from "Real Analysis: Foundations and Functions of One Variable" applies not only to real numbers but also to higher dimensions, specifically in R^k. Participants confirm that the theorem's principles extend to R^n, indicating its broader applicability in analysis. This extension is significant for understanding limits and subsequences in multi-dimensional spaces. The discussion emphasizes the importance of recognizing such theorems in various mathematical contexts. Overall, Theorem 5.2 is relevant for both R and R^n.
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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 $$\mathbb{R}^n$$?Peter
 
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Yes, Theorem 5.2 holds in $\Bbb R^k$, not just $\Bbb R$.
 

Thread 'About the definition of topological manifold using closed sets'

We all know the definition of n-dimensional topological manifold uses open sets and homeomorphisms onto the image as open set in ##\mathbb R^n##. It should be possible to reformulate the definition of n-dimensional topological manifold using closed sets on the manifold's topology and on ##\mathbb R^n## ? I'm positive for this. Perhaps the definition of smooth manifold would be problematic, though.
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