Analysis - compactness and sequentially compactness

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SUMMARY

The discussion centers on the concepts of compactness and sequential compactness within metric spaces, specifically referencing the real numbers (R) and theorems such as Bolzano-Weierstrass and Heine-Borel. It is established that in R^n, compactness is equivalent to being closed and bounded, while sequential compactness implies that every sequence has a converging subsequence within the set. The confusion arises from considering R as sequentially compact despite it being unbounded, leading to questions about the convergence of sequences like {n} = {1, 2, 3, 4,...}.

PREREQUISITES
  • Understanding of metric spaces
  • Familiarity with the Bolzano-Weierstrass theorem
  • Knowledge of the Heine-Borel theorem
  • Concept of convergence in sequences
NEXT STEPS
  • Study the implications of the Bolzano-Weierstrass theorem in different metric spaces
  • Explore the Heine-Borel theorem in detail, particularly in R^n
  • Investigate examples of non-compact sets and their properties
  • Learn about the differences between compactness and sequential compactness in various contexts
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Mathematicians, students of analysis, and anyone studying topology or metric spaces will benefit from this discussion.

quasar987
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quasar987 said:
1. Homework Statement
According to the dfn, a subset A of a metric space is sequentially compact is every sequence in A has a subsequence that converges to a point in A.

An example of a sequentially compact set that comes to mind is R itself.

Then, Bolzano-Weierstrass's thm says that sequentially compactness and compactness are equivalent.

Finally, Heine-Borel's thm says that in R^n, compactness and closed+bounded are equivalent.

Thus, in R^n, closed+bounded and sequentially compactness are equivalent. But R is not bounded. What's going on?
Why would R "come to mind" as an example of a sequentially compact set? In particular what do you say the sequence {n}= {1, 2, 3, 4,...} converges to?
 

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