Let {a_n}_{ninmathbb{N}} be a bounded sequence that does not converge. By the Bolzano-Weierstrass Theorem, {a_n}_{ninmathbb{N}} has a convergent subsequence that converges to some real number L. (a) Prove that there exists a subsequence of {a_n}_{ninmathbb{N}} that converges to a number M in R where M ≠ L. (b) Must there exist a third subsequence that converges to a number P where P ≠ M and P ≠ L? Justify with a proof or counterexample.
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In this project, you will prove that every sequence has a monotone subsequence and then, without using the Bolzano-Weierstrass Theorem, prove that every Cauchy sequence converges. THEOREM Every sequence has a monotone subsequence. Proof 1. For a sequence {a_n}_{n=1}^{∞}, prove that exactly one of the following statements is true: i. There is N such that no term of the sequence {a_{N+k}}_{k=1}^{∞} is an upper bound of that sequence. ii. For each N, there is a term of the sequence {a_{N+k}}_{k=1}^{∞} which is an upper bound for that sequence. 2. Prove that {a_n}_{n=1}^{∞} has a monotone subsequence as follows: i. If (i) in item 1 is true, construct an increasing sequence that is a subsequence of {a_n}_{n=1}^{∞}. ii. If (ii) of item 1 is true, construct a decreasing sequence that is a subsequence of {a_n}_{n=1}^{∞}. THEOREM Every Cauchy sequence is convergent. Proof Use the theorem above.
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Theorem 6.1.2 Let (an) be a sequence in a metric space X. 1. If (an) converges, then it is a Cauchy sequence. 2. If (an) is a Cauchy sequence, then every subsequence of (an) is a Cauchy sequence. 3. If (an) is a Cauchy sequence, then it must be bounded. 4. If (an) is a Cauchy sequence that has a convergent subsequence, then (an) itself must converge. Proof: The proof is Problem 1 at the end of this section. Problems 6.1 1. Prove Theorem 6.1.2, which lays out the basic characteristics of Cauchy sequences. (Hint: In the proof of the last part, you may find a use for Theorem 0.4.6.)
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Let (xn)n=1 be a sequence in a metric space (X, d). (a) Suppose (xn)n=1 is a Cauchy sequence. Show that {xn | 1 <= n < inf} is a bounded set. (b) Suppose (xn)n=1 is a Cauchy sequence, and that there is a subsequence (xnj)j=1 which converges to some x in X. Show that also lim n->inf xn = x. (c) We say that x in X is a limit point of the sequence (xn)n=1 if for every e > 0, for every N in N, there exists n >= N such that d(xn, x) < e. Show that x is a limit point of (xn)n=1 if and only if there exists a subsequence (xnj)j=1 which converges to x.
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