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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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