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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