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\title{MTH435 - Homework 6}



\begin{document}
 \maketitle
 
 Solutions to the questions in Section B should be submitted by the start of class on 11/7/18.
 
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 \section{Warm-up Questions}
 
 \begin{que} Prove the following assertions which strengthen the Root and Ratio tests.
 \begin{enumerate}
  \item If $\limsup |x_n|^{\frac{1}{n}} = r$ and $r<1$ then $\sum x_n$ converges absolutely.
  \item If $\limsup \left| \frac{x_{n+1}}{x_n} \right| = r$ and $r < 1$ then $\sum x_n$ converges absolutely.
 \end{enumerate}
 \end{que}
 
 \begin{que}
 Discuss the convergence or divergence of $\sum x_n$ for the following sequences.
 \begin{enumerate}
  \item $x_n = \frac{\sqrt{n+1} - \sqrt{n}}{n}$.
  \item $x_n = \frac{1}{n \log n}$.
  \item $x_n = \frac{n}{2^n}$.
  \item $x_n = \frac{n^n}{e^n}$.
  \item $x_{2n-1} = \frac{1}{(2n-1)^2}$ and $x_{2n} = \frac{1}{(2n)^3}$.
  \item $x_n = \frac{n!}{n^n}$.
 \end{enumerate}

\end{que}
 
 \begin{que}
  Prove the following are metric spaces.
  \begin{enumerate}
   \item $d(x,y) = \frac{|x-y|}{1+|x-y|}$ on $\R$.
   \item $d(x,y) = \min\{1,|x-y|\}$ on $\R$.
   \item $d((x_1,y_1),(x_2,y_2)) = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$ on $\R^2$.
   \item $d(x,y) = 1$ if $x \neq y$ and $d(x,y) = 0$ if $x = y$ on an arbitrary non-empty set $X$.
   \item $d(x,y) = |x-y|$ on $\Q$.
   \item $d((x_1,y_1),(x_2,y_2),y) = \max \{ |x_1 - x_2|, |y_1 - y_2| \}$ on $\R^2$.
  \end{enumerate}

 \end{que}

 \begin{que}
  Show that the following are \emph{not} metric spaces.
  \begin{enumerate}
   \item $d(x,y) = |x-y|^2$ on $\R$.
   \item $d((x_n),(y_n)) = | \lim_{n \to \infty} (x_n - y_n) |$ on the set of convergent sequences.
   \item $d((x_n),(y_n)) = \sup \{ |x_n - y_n| \mid n \in \N \}$ on the set of sequences of real numbers.
   \item $d(X,Y) = \inf\{ |x-y| \mid x \in X, y \in Y \}$ on the set of non-empty subsets of $\R$.
  \end{enumerate}

 \end{que}

 
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 \section{Submitted Questions}
 
 \begin{que}
  Suppose $a_n > 0$ for all $n$. By using the following steps (or otherwise, if you find a better method!), prove that if $\sum \frac{a_n}{1 + a_n}$ converges then $\sum a_n$ converges.
   \begin{enumerate}
   \item Show that if $\sum \frac{a_n}{1 + a_n}$ converges then there exists $K$ such that if $n \geq K$ then $0 < a_n < 1$.
   \item Deduce that $n \geq K$ implies $\frac{a_n}{2} \leq \frac{a_n}{1+a_n}$.
   \item Prove that $\sum a_n$ converges.
   \end{enumerate}

 \end{que}

 
 
 \begin{que}
  Let $X$ be the set of bounded sequences of real numbers. Define $d \colon X \times X \to \R$ by $d((x_n),(y_n)) = \sup \{ |x_n - y_n| \mid n \in \N \}$. Prove that $(X,d)$ is a metric space.
 \end{que}

 
 
 
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 \section{Challenge Questions}
 
 \begin{que}
Let $X$ be the set of sequences $(x_n)$ for which the associated series $\sum x_n$ is absolutely convergent and define $d \colon X \times X \to \R$ by $d((x_n),(y_n)) = \sum |x_n - y_n|$. Prove that $(X,d)$ is a metric space.
 \end{que}
 
\begin{que}
  Let $X$ be a space with a semi-metric $d$ (that is, $d$ satisfies all the properties of a metric except it is not necessarily true that $d(x,y)=0$ implies $x=y$.)
  \begin{itemize}
   \item[(i)] Prove that the relation $\sim$ on $X$ defined by $x \sim y$ if and only if $d(x,y) = 0$ is an equivalence relation on $X$.
   \item[(ii)] Let $X/ {\sim}$ be the set of equivalence classes of $\sim$. Prove that $(X/{\sim},d')$ is a metric space, where $d'([x],[y]) = d(x,y)$ (here, $[x]$ denotes the equivalence class of $x$ under $\sim$ - you will have to check $d'$ is well-defined).
  \end{itemize}

 \end{que}
 
 
 
 
 
 \end{document}
 
 
 
 
 
 
 
 
 
 
