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\centerline{Math 101}
\centerline{Homework 13}

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 Exercise 9.10 and the following.
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\noindent 1)     Suppose that $s_n\rightarrow \infty$ and $t_n\rightarrow a$ with $a<0$.  Prove that $s_nt_n\rightarrow -\infty$.

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 \noindent 2)   Let $\{x_n\}$ and $\{y_n\}$ be sequences in $\mathbb{R}-\{0\}$.  Suppose that $\frac{x_n}{y_n}\rightarrow 1$, and that one of the sequences $\{x_n\}$ or $\{y_n\}$ is bounded.  Prove that $x_n-y_n\rightarrow 0$.  Give an example where $\{x_n\}$ and $\{y_n\}$ are both unbounded, $\frac{x_n}{y_n}\rightarrow 1$, and $x_n-y_n\not\rightarrow 0$.  You don't have to prove all claims about your example.\medskip
 
 \noindent 3) Suppose that $s_n\rightarrow -\infty$ and $k<0$.  Prove that $ks_n\rightarrow \infty$.
 
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 \centerline{Homework 14}

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Exercise 9.11.  Note in this exercise it says $\lim t_n > -\infty$.   This means either $\{t_n\}$ converges or $t_n\to \infty$.

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Also, do the following.

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 \noindent 1)  Let $x_n\rightarrow l$.  Let $\{y_n\}$ be a sequence obtained by rearranging the order of the terms of $\{x_n\}$.  Prove that $y_n\rightarrow l$.  

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  \noindent 2)   Let $a=\mathrm{lub}\{a_n\}$ and $b=\mathrm{lub}\{b_n\}$.  Prove that $\mathrm{lub}\{a_n+b_n\}\leq a + b$.  Give an example where $\mathrm{lub}\{a_n+b_n\}< a + b$.  You don't have to prove all claims about your example.

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  \noindent 3)  Suppose that $S$ is a set of real numbers which is not bounded above.  Prove that there exists a sequence $\{x_n\}$ contained in $S$, such that $x_n\rightarrow \infty$.  
  
  
 
  
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\centerline{Homework 15}

  
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 \noindent  Exercises  10.2, 10.5, and the following
 
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  \noindent 1)     Let $l\in \mathbb{R}$.  Prove that there exists a sequence $\{x_n\}$ of rationals and a sequence $\{y_n\}$ of irrationals, such that $x_n\rightarrow l$ and $y_n\rightarrow l$. 
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  \noindent 2) Let $S$ be a bounded nonempty subset of real numbers and suppose that $\mathrm{lub}(S)\not \in S$.  Prove that there is an increasing sequence $\{s_n\}$ of points in $S$ such that $\lim s_n=\mathrm{lub}(S)$.
  
  
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\centerline{Homework 16}

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\noindent  Note:   When doing this homework you may not assume that Cauchy sequences converge.

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  \noindent 1) Suppose that $\{x_n\}$ and $\{y_n\}$ are Cauchy sequences.  Prove that $\{x_n+y_n\}$ is Cauchy.   
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   \noindent 2)  Suppose that $\{x_n\}$ is a sequence of integers which is Cauchy.  Prove that there exists an $N \in \mathbb{N}$ such that for every $n>N$ $x_n=x_N$.    

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 \noindent 3) Prove that the sequence $\{x_n\}=\{\sqrt{n}\}$ is not Cauchy, and prove that for every $\varepsilon>0$ there is an $N \in \mathbb{N}$ such that if $n>N$ then $|x_{n+1}-x_n|<\varepsilon$.    
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 \noindent 4) Let $\{x_n\}$ and $\{y_n\}$ be Cauchy sequences.  Prove that $\{x_ny_n\}$ is Cauchy.  


 

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