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The definition of x being a limit point of a set A is "x € R such that V€ > 0, N₁(x)^A\ {x} ‡ Ø”. Write down statement of A does not have any limit point. Using the concept of open covers and explicitly avoiding the Bolzano-Weierstrass Theorem, prove that if A is a bounded infinite set, then A has a limit point. You may prove by contradiction and use Heine-Borel Theorem.

The definition of x being a limit point of a set A is "x € R such that V€ > 0, N₁(x)^A\ {x} ‡ Ø”. Write down statement of A does not have any limit point. Using the concept of open covers and explicitly avoiding the Bolzano-Weierstrass Theorem, prove that if A is a bounded infinite set, then A has a limit point. You may prove by contradiction and use Heine-Borel Theorem.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
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Section3.CR: Chapter 3 Review
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3. (a) The definition of x being a limit point of a set A is "x ER such that Ve > 0, N₁(x)^A\ {x} ‡ Ø”. Write down statement of A does not have any limit point. (b) Using the concept of open covers and explicitly avoiding the Bolzano-Weierstrass Theorem, prove that if A is a bounded infinite set, then A has a limit point. You may prove by contradiction and use Heine-Borel Theorem.
Transcribed Image Text:3. (a) The definition of x being a limit point of a set A is "x ER such that Ve > 0, N₁(x)^A\ {x} ‡ Ø”. Write down statement of A does not have any limit point. (b) Using the concept of open covers and explicitly avoiding the Bolzano-Weierstrass Theorem, prove that if A is a bounded infinite set, then A has a limit point. You may prove by contradiction and use Heine-Borel Theorem.
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