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%%% exercise set 6
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\begin{document}

\begin{center}
Math 445,  Fall 2009 \hfill  Exercise Set \#6 \hfill Turn in: October 16  
\end{center}
 
 \bigskip
 
  
 {\bf 1.}   [\#1, page 93] ~         Prove that the space $X = \cB(\mN, \mR)$ of bounded sequences with the ``sup norm''  is complete. [This is Example 7, page 119 of Appendix 1 in the text.] 
 
 The metric is defined by, for
  $x = \{x_n\} \in \cB(\mN, \mR)$ and $y = \{y_n\} \in \cB(\mN, \mR)$
 $$D(x,y) ~ = ~ \sup_{n \in \mN} ~ |x_n - y_n| $$
     
 \bigskip
 
 
 {\bf 2.}   [\#5, page 93] ~   If every countable closed subset of a metric space $M$ is complete, prove that $M$ is complete. 
     
 \bigskip
 
 
 {\bf 3.}   [\#6, page 93] ~    Let $\{M, d\}$ be a metric space.    If for every $u \in M$ and $\epsilon > 0$, the closed ball  $D(u,\epsilon) = \{ y \in M \mid d(u,y) \leq \epsilon\}$  is complete, prove that  $\{M, d\}$ is complete. 
     
 \bigskip
 
 
 {\bf 4.}   [\#11, page 93] ~      In the space of real numbers $\mR$, give an example of a descending sequence of non-empty closed sets with empty intersection.   That is, find $F_1 \supset F_2 \supset \cdots $ where each $F_n \subset \mR$ is closed, and $\ds \bigcap_{n=1}^{\infty} ~ F_n ~ = ~ \emptyset$.
     
 \bigskip
 
 
 {\bf 5.}   [\#12, page 93] ~         The following functions are continuous from the real numbers to the real numbers. Which are \emph{uniformly} continuous?
    
    a) $ f(x) = x^2$
    
    b)  $g(x) = |x|$
    
    c) $\ds h(x) = \frac{1}{1 + x^2}$
    
 \bigskip
 
  {\bf 6.}   [\#13, page 93] ~         Let $\mR$ be the real line with the standard metric, $d(x,y) = |x-y|$. Let $f \colon \mR \to \mR$ be a function which has derivative $f'(x)$ for  every $x \in \mR$.  Assume  there exists $K \geq 0$ such that $| f'(x)| \leq K$ for all $x \in \mR$. Prove that $f$ is uniformly continuous. [Hint: Mean Value Theorem]
  
     
  
 
 

 
  
 \vfill
 
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