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\begin{document}
\begin{center}
\Large{Exercises for Module 5: Metric Spaces III}
\end{center}

1. Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces and let $f:X\to Y$. Prove that
$$f \text{ is Lipschitz continuous } \Rightarrow f \text{ is uniformly continuous } \Rightarrow f \text{ is continuous}.$$
Provide examples to show that the other directions do not hold.

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2. Show that the function $f(x) = \frac{1}{2} \left(x + \frac{5}{x} \right)$ has a unique fixed point on $(0,\infty)$. What is it? (Hint: you will have to restrict the interval.)
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3.  Prove the following: If two metrics are strongly equivalent then they are equivalent. 

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4. Let $(X,d)$ be a metric space and $\{A_i\}_{i\in I}$ be a collection of subsets of $X$. Show that 
    \begin{equation*}
        \bigcup_{i\in I}\overline{A_i} \subseteq \overline{\bigcup_{i\in I}A_i}.
    \end{equation*}
    Show that if the collection is finite, the two sets are equal. 

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5. Let $(X,d)$ be a metric space and $\{A_i\}_{i\in I}$ be a collection of subsets of $X$. Prove that 
    \begin{equation*}
        \overline{\bigcap_{i\in I}A_i} \subseteq \bigcap_{i\in I}\overline{A_i}.
    \end{equation*}
    Find a counterexample that shows that equality is not necessarily the case.
    
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6.  Let $(X,d)$ be a metric space and $A\subseteq X$ be dense. Show that if $A\subseteq B\subseteq X$, then $B$ is dense as well. 

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