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\title{ Math 3210-4,  HW IV \\ Due Wednesday Oct 30 }
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1)  Let $f: X \rightarrow Y$ be a continuous map of metric spaces.  Let $E\subset X$. 
Prove that $f(\overline E) \subseteq \overline{f(E)}$. Give an example where the inclusion is proper. 



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2) Let  $F_1\supseteq F_2 \supseteq \ldots $ be a  descending sequence of non-empty compact sets in a metric space $X$. 
Prove that $\cap_{n=1}^{\infty} F_n$ is not empty. 
Give an example of a descending sequence of closed sets such that the intersection is empty. 



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3) Let $f: X \rightarrow Y$ be a uniformly continuous map of metric spaces.  If $x_n$ is a Cauchy sequence in $X$, prove that 
$f(x_n)$ is a Cauchy sequence in $Y$. Give an example of a continuous function such that this fails. 


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4) Let $f:[0,1] \rightarrow [0,1]$  be a continuous function. Prove that there exists $x\in [0,1]$ such that $f(x)=x$. 


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5) Let $X$ be a metric space, and fix a non-empty subset $E\subset X$. The distance from $x\in X$ to $E$ is defined by 
\[ 
d(x,E)= \mathrm{inf}_{y\in E} d(x,y).
\]  
Prove that $d(x,E)=0$ if and only if $x\in \overline E$. Prove that 
\[ 
|d(x,E)-d(y,E) | \leq d(x,y) 
\] 
i.e. $x\mapsto d(x, E)$ is a uniformly continuous function. 




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