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\title{MTH435 - Homework 5}



\begin{document}
 \maketitle
 
 Solutions to the questions in Section B should be submitted by the start of class on 10/17/18.
 
 
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 \section{Warm-up Questions}
 
 \begin{que}
  Using the definition, decide whether or not the following sequences are Cauchy: $x_n =  \frac{1}{n(n+1)}$, $y_n = \sqrt{n}$ and $z_n = \frac{n-1}{n}$.
 \end{que}
 
 
 \begin{que}
  Let $(x_n)$ and $(y_n)$ be Cauchy sequences. \emph{Directly from the definition\dots}
  \begin{enumerate}
   \item Prove that $(x_n + y_n)$ is Cauchy.
   \item Prove that $(x_n y_n)$ is Cauchy.
   \item Prove that $|x_n - y_n|$ converges.
   \item Prove that if $x_n \in \Z$ for all $n \in \N$ then $(x_n)$ is eventually constant.
  \end{enumerate}
 
 \end{que}
 
 \begin{que}
 Prove or disprove the following for a sequence $(x_n)$.
 \begin{enumerate}
 \item $(x_n)$ is Cauchy if and only if $ \lim_{n \to \infty} |x_{n+1} - x_n| = 0$.
 \item $(x_n)$ is contractive if and only if it is Cauchy.
 \end{enumerate}
 \end{que}
 
 \begin{que}
  Let $x_1 = 5$ and $x_{n+1} = \frac{1}{2+x_n}$.
  \begin{enumerate}
   \item Prove that $(x_n)$ is contractive.
   \item Find the limit of $(x_n)$.
  \end{enumerate}

 \end{que}

 
\begin{que}
  Let $f \colon \R \to \R$ be defined by $f(x) = 3 - \frac{x}{2}$. Define a sequence by $x_1 = 10$ and $x_{n+1} = f(x_n)$.
  \begin{enumerate}
   \item Prove that $x_n$ converges.
   \item Prove that $x = \lim_{n \to \infty} x_n$ is the unique real number satisfying $f(x) = x$.
   \item Prove that the value of the limit $x$ does not depend on the initial choice of $x_1$.
  \end{enumerate}

 \end{que}
 
 
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 \section{Submitted Questions}
 
 \begin{que}
  Suppose $(x_n)$ is a bounded, increasing sequence. Prove directly (without referring to the Monotone Convergence Theorem or the Cauchy Convergence Criterion) that $(x_n)$ is Cauchy.
 \end{que}
 
 \begin{que}
  Let $x_1 = 4$ and $x_{n+1} = 2 + \frac{1}{x_n}$. Prove that $(x_n)$ converges and find the limit.
 \end{que}


 

 
 
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 \section{Challenge Questions}
 
 \begin{que} \emph{Contraction Mapping Theorem}.
  Suppose $f \colon \R \to \R$ is \emph{contractive}. That is, there exists $0 \leq \ell < 1$ such that
  $|f(x) - f(y)| \leq \ell |x - y|$
  for all $x,y \in \R$. Let $x_1 \in \R$ and define a recursive sequence by $x_{n+1} = f(x_n)$. You may assume $\lim_{n \to \infty} f(x_n) = f(\lim_{n \to \infty} x_n)$. 
  \begin{enumerate}
   \item Prove that there exists $x \in \R$ such that $x_n \to x$.
   \item Prove that $x$ must satisfy $f(x) = x$, and is the unique solution to this equation.
   \item Deduce that $x$ is independent of the initial choice of $x_1$.
  \end{enumerate}
 \end{que}
 
 
 \begin{que}
 \emph{Completion of $\Q$}. Consider the set $\mathcal{C}(\Q)$ of Cauchy sequences with all terms in $\Q$. Define a relation $\sim$ on $\mathcal{C}(\Q)$ by $(x_n) \sim (y_n)$ if and only if $(x_n - y_n) \to 0$.
 \begin{enumerate}
  \item Show that $\sim$ is an equivalence relation on $\mathcal{C}(\Q)$.
  \item Let $X^\ast$ be the set of equivalence classes of $\sim$. Let $X, Y \in X^\ast$ and for $(x_n) \in X$, $y_n \in Y$, define $D \colon x^\ast \times X^\ast \to \R$ by $D(X,y_n) = \lim_{n \to \infty} |x_n - y_n|$. Prove that $D$ is well-defined; that is, it does not depend on the chosen representatives $x_n$ and $y_n$.
  \item Prove the following.
  \begin{enumerate}
  \item $D(X,Y) = 0$ if and only if $X = Y$.
  \item $D(X,Y) = D(Y,X)$ for all $X,Y \in X^\ast$.
  \item $D(X,Y) \leq D(X,Z) + D(Z,Y)$ for all $X,Y,Z \in X^\ast$.  
  \end{enumerate}
  \item (\emph{Hard}) Prove that $X^\ast$ is complete; that is, all Cauchy sequences converge in $X^\ast$. (Cauchy and convergence are defined analogously to in the real numbers, but with $D$ instead of the absolute value used to measure distance).
 \end{enumerate}

 \end{que}

 
 
 
 
 
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