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 n;. (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 n

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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, Tn has a
smallest element. Prove that there exists an increasing subsequence Inj.
has
(ii) Suppose the condition above fails, so that there exists NEN such that TN
no smallest element. Prove that there exists a decreasing subsequence n;.
b) Give an easy alternative proof of the Bolzano-Weierstrass Theorem.
om/terms-and-conditions) on Wil
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, Tn has a smallest element. Prove that there exists an increasing subsequence Inj. has (ii) Suppose the condition above fails, so that there exists NEN such that TN no smallest element. Prove that there exists a decreasing subsequence n;. b) Give an easy alternative proof of the Bolzano-Weierstrass Theorem. om/terms-and-conditions) on Wil
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