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\centerline{Math 101}
\centerline{Homework 19}
Exercise 11.6
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\noindent 1) Let $\{x_n\}$ be an increasing sequence, and suppose $\{x_{n_k}\}$ is a subsequence which converges to $l$. Prove that $x_n\rightarrow l$.
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\noindent 2) Let $a\in \mathbb{R}$. Let $\{x_n\}$ have subsequences $\{x_{n_k}\}$ and $\{x_{m_j}\}$, such that every term of $\{x_n\}$ is either equal to $a$ or is contained in one of these two subsequences. Suppose that both subsequences converge to $a$. Prove that $x_n\rightarrow a$.
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\noindent 3) Give examples of sequences and subsequences which satisfy the following. You don't have to prove all claims about your examples.
a) $\{x_n\}$ is not increasing, but $\{x_n\}$ has an increasing subsequence.
b) $\{x_n\}$ is unbounded, but $\{x_n\}$ has a bounded subsequence.
c) A sequence of integers, $\{x_n\}$ which diverges, but which has infinitely many distinct subsequential limits.
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\centerline{Homework 20}
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\noindent 1) Imitate the proof of the MST to prove that every sequence either has a decreasing subsequence or a non-decreasing subsequence.
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\noindent 2) Let $a$ be a limit point of a sequence $\{x_n\}$. Let $\{y_n\}$ be a sequence obtained by rearranging the order of the terms of $\{x_n\}$ and adding a finite or infinite number of additional terms. Prove that $a$ is a limit point of $\{y_n\}$.
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\noindent 3) Suppose that $x_n\not\rightarrow\infty$. Prove that either $\{x_n\}$ has a subsequence which converges or $\{x_n\}$ has a subsequence which diverges to $-\infty$.
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\noindent 4) Prove or disprove each of the following statements. You can disprove something with a counterexample, you don't have to prove your claims about the counterexample.
a) Every sequence has an increasing subsequence.
b) Every sequence has a bounded subsequence.
c) Every unbounded sequence has a subsequence which either diverges to $\infty$ or to $-\infty$.
d) If a sequence has a greatest term, then every subsequence has a greatest term.
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