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\title{MTH435 - Homework 8}



\begin{document}
 \maketitle
 
 Solutions to the questions in Section B should be submitted by the start of class on 11/14/18. In this homework, the metrics $d_1$, $d_2$ and $d_\infty$ on $\R^2$ are defined as follows.
   \begin{align*}
   d_1((x_1,y_1),(x_2,y_2)) &= |x_1 - x_2| + |y_1 - y_2| \\
   d_2((x_1,y_1),(x_2,y_2)) &= \sqrt{|x_1 - x_2|^2 + |y_1 - y_2|^2} \\
   d_\infty((x_1,y_1),(x_2,y_2)) &= \max \{ |x_1 - x_2|, |y_1 - y_2| \}
  \end{align*}
 On any non-empty set $X$, the discrete metric $\delta$ is defined as
 \[
  \delta(x,y) = \begin{cases}
                 0& \quad \text{if $x=y$}, \\
                 1& \quad \text{if $x \neq y$}.
                \end{cases}
 \]
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \section{Warm-up Questions}
 
 
 \begin{que}
  Draw $V_1((0,0))$ in $\R^2$ for the metrics $d_1$, $d_2$, $d_\infty$ and $\delta$.
 \end{que}
 
 \begin{que}
  Let $(X,d)$ be a metric space.
  \begin{enumerate}
   \item Show that if $x,y \in X$, $r \in \R$ and $d(x,y) > 2r$ then $V_r(x)$ and $V_r(y)$ are disjoint.
   \item Show that if $x \neq y$ in $X$ then there exist neighbourhoods $U$ of $x$ and $V$ of $y$ such that $U \cap V = \varnothing.$\footnote{This is called the \emph{Hausdorff} property. You can remember the definition by noting that it says $x$ and $y$ can be ``housed off'' from each other.} 
   \item Deduce that, for any metric space $X$ containing more than one point, it is not possible that $X$ and $\varnothing$ are the only open subsets of $X$.
  \end{enumerate}

 \end{que}

 
 \begin{que}
  Let $X = [0,1]$ with the usual absolute value as a metric. Is the set $U = \left[0,\frac{1}{2} \right)$ open? Is it closed?
 \end{que}

 \begin{que}
  Prove that if $X$ is a discrete metric space and $(x_n)$ is a Cauchy sequence then $(x_n)$ is eventually constant.
 \end{que}

 \begin{que}
  Prove that the metrics $d_1, d_2$ and $d_\infty$ on $\R^2$ are topologically equivalent. In fact, prove that
   \[
   \frac{1}{2}d_1(a,b) \leq \frac{1}{\sqrt{2}}d_2(a,b) \leq d_\infty(a,b) \leq d_2(a,b) \leq d_1(a,b). 
  \]
for all $a = (x_1,y_1)$, $b=(x_2,y_2)$ in $\R^2$.
 \end{que}

 
 \begin{que}
  Using the notation from class, prove that if $d$ and $d'$ are Lipschitz equivalent metrics, then there exists $K > 0$ such that for all $x \in X$ we have
  \[
   V'_{\frac{\eps}{K}}(x) \subseteq V_\eps(x) \subseteq V'_{K \eps}(x).
  \]

 \end{que}

 \begin{que}
  Suppose that $d$ and $d'$ are equivalent metrics on a set $X$. Prove that $(X,d)$ is complete if and only if $(X,d')$ is complete, 
 \end{que}

 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \section{Submitted Questions}
 
  \begin{que} 
 Prove that $\R^2$ equipped with $d_\infty$ is a complete metric space.
 \end{que}
 
 \begin{que}
 Consider the metrics $d(x,y) = |x-y|$ and $d'(x,y) = \min\{1,|x-y|\}$.
 \begin{enumerate}
  \item Prove that $d$ and $d'$ are topologically equivalent.
  \item Prove that $d$ and $d'$ are not Lipschitz equivalent.
 \end{enumerate}

\end{que}

 
 
 
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \section{Challenge Questions}
 
 \begin{que}
  It was shown in class that one of the consequences of the Completeness Property of $\R$ is that all Cauchy sequences converge. In this question, we will show that if we assume all Cauchy sequences converge \emph{and} $\R$ has the Archimedean Property then every non-empty set in $\R$ which is bounded above has a least upper bound. This will prove that our definition of completeness of a metric space is equivalent to our original definition of completeness in $\R$.\footnote{Unless you assume $\R$ does not satisfy the Archimedean property, which leads to topics such as Non-standard Analysis and $p$-adic Analysis.}
  \begin{itemize}
   \item[(i)] Prove that if $(x_n)$ is an increasing sequence which is bounded above, then it is Cauchy. Deduce that all bounded monotonic sequences converge.
   \item[(ii)]  Let $A$ be a non-empty set which is bounded above. Construct a Cauchy sequence $(a_n)$ in $A$ as follows.
   \begin{itemize}
    \item Let $a_1 \in A$ and $u_1$ be an upper bound of $A$. Define $p_1 = \frac{1}{2}(a_1 + u_1)$.
    \item If $p_1$ is an upper bound of $A$, define $a_2 = a_1$ and $u_2 = p_1$.
    \item If $p_1$ is not an upper bound of $A$, choose $a_2 \in A$ such that $a_2 > p_1$. Then define $u_2 = u_1$.
    \item Continue inductively, defining $p_{k+1} = \frac{1}{2}(a_k + u_k)$. Deduce that $(a_k)$ are $(u_k)$ are monotonic sequences and thus converge.
   \end{itemize}
   \item[(iii)] Prove that $\sup A = \lim_{n \to \infty} a_n = \lim_{n \to \infty} u_n$.
  \end{itemize}

 \end{que}
 
 
 
 
 
 \end{document}
 
 
 
 
 
 
 
 
 
 
