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Bolzano-Weierstrass theorem is
- a)every bounded sequence has a limit point
- b)the set of limit points of a bounded sequence is bounded
- c)an unbounded sequence may or may not have a limit point
- d)None of the above
Correct answer is option 'A'. Can you explain this answer?
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Bolzano-Weierstrass theorem isa)every bounded sequence has a limit poi...
Understanding the Bolzano-Weierstrass Theorem
The Bolzano-Weierstrass theorem is a fundamental result in real analysis, specifically concerning sequences in Euclidean spaces.
Key Statement
- Every bounded sequence in R^n has a convergent subsequence.
Explanation of Option A
- This option states that every bounded sequence has a limit point.
- A *bounded sequence* is one that is contained within some fixed interval or region in space.
- Since a bounded sequence cannot "escape" to infinity, it must cluster around certain values.
- The theorem assures us that among these values, there exists at least one limit point, which is a point where the sequence gets arbitrarily close infinitely often.
Why Other Options Are Incorrect
- Option B: The set of limit points of a bounded sequence is bounded.
- This is true but does not directly state the core of the theorem. The theorem asserts the existence of at least one limit point, not the boundedness of the set of limit points.
- Option C: An unbounded sequence may or may not have a limit point.
- This is also true but does not relate to the theorem. The Bolzano-Weierstrass theorem specifically addresses bounded sequences.
- Option D: None of the above.
- This is incorrect since option A accurately reflects the theorem.
Conclusion
The Bolzano-Weierstrass theorem emphasizes the significance of boundedness in sequences, guaranteeing that every bounded sequence has at least one limit point, which is essential in analysis for understanding convergence.
The Bolzano-Weierstrass theorem is a fundamental result in real analysis, specifically concerning sequences in Euclidean spaces.
Key Statement
- Every bounded sequence in R^n has a convergent subsequence.
Explanation of Option A
- This option states that every bounded sequence has a limit point.
- A *bounded sequence* is one that is contained within some fixed interval or region in space.
- Since a bounded sequence cannot "escape" to infinity, it must cluster around certain values.
- The theorem assures us that among these values, there exists at least one limit point, which is a point where the sequence gets arbitrarily close infinitely often.
Why Other Options Are Incorrect
- Option B: The set of limit points of a bounded sequence is bounded.
- This is true but does not directly state the core of the theorem. The theorem asserts the existence of at least one limit point, not the boundedness of the set of limit points.
- Option C: An unbounded sequence may or may not have a limit point.
- This is also true but does not relate to the theorem. The Bolzano-Weierstrass theorem specifically addresses bounded sequences.
- Option D: None of the above.
- This is incorrect since option A accurately reflects the theorem.
Conclusion
The Bolzano-Weierstrass theorem emphasizes the significance of boundedness in sequences, guaranteeing that every bounded sequence has at least one limit point, which is essential in analysis for understanding convergence.
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