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b) Give an easy alternative proof of the Bolzano-Weierstrass Theorem.

b) Give an easy alternative proof of the Bolzano-Weierstrass Theorem.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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only part b, please

1.64 The following questions provide an easy, alternative proof of the Bolzano- Weierstrass Theorem. a) Use the following steps to prove that every sequence xn of real numbers has a monotone subsequence. Denote the nth tail of the sequence by Tn = {xj|j≥n}. (i) Suppose the following special condition is satisfied: For each n E N, T has a smallest element. Prove that there exists an increasing subsequence anz. (ii) Suppose the condition above fails, so that there exists NEN such that TN has no smallest element. Prove that there exists a decreasing subsequence xnj. b) Give an easy alternative proof of the Bolzano-Weierstrass Theorem.
Transcribed Image Text:1.64 The following questions provide an easy, alternative proof of the Bolzano- Weierstrass Theorem. a) Use the following steps to prove that every sequence xn of real numbers has a monotone subsequence. Denote the nth tail of the sequence by Tn = {xj|j≥n}. (i) Suppose the following special condition is satisfied: For each n E N, T has a smallest element. Prove that there exists an increasing subsequence anz. (ii) Suppose the condition above fails, so that there exists NEN such that TN has no smallest element. Prove that there exists a decreasing subsequence xnj. b) Give an easy alternative proof of the Bolzano-Weierstrass Theorem.
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