Proof of Bolzano-Weierstrass theorem using Axiom of Completeness.

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SUMMARY

The discussion focuses on the proof of the Bolzano-Weierstrass theorem utilizing the Axiom of Completeness. Participants analyze the sequence defined by a_n = (-1)^n/n, illustrating its convergence behavior. The set S is identified as containing all x ≤ 0, while x > 0 is excluded, highlighting the theorem's implications regarding bounded sequences in real analysis.

PREREQUISITES
  • Understanding of the Bolzano-Weierstrass theorem
  • Familiarity with the Axiom of Completeness in real analysis
  • Knowledge of sequences and their convergence
  • Basic graphing skills to visualize sequences
NEXT STEPS
  • Study the implications of the Axiom of Completeness on real numbers
  • Explore examples of bounded sequences and their limits
  • Learn about the applications of the Bolzano-Weierstrass theorem in functional analysis
  • Investigate graphical methods for analyzing convergence of sequences
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Mathematics students, educators, and researchers interested in real analysis, particularly those studying convergence and the properties of bounded sequences.

kindlychung
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This is an analysis exercise, I don't get the definition of S. Could anyone explain, please?
See the attached pic.
 

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It might help you to look at an example. Let an = (-1)n/n. Plot a few of these: -1, 1/2, -1/3, 1/4, ...

Now what x would be in your set S? Can you see any x ≤ 0 is in S but any x > 0 isn't?
 

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