\documentclass[11pt]{amsart}
\usepackage{amscd,amssymb}
\theoremstyle{plain}
\newtheorem{Thm}[equation]{Theorem}
\newtheorem{Cor}[equation]{Corollary}
\newtheorem{Prop}[equation]{Proposition}
\newtheorem{Lem}[equation]{Lemma}
\newtheorem{Def}[equation]{Definition}
\numberwithin{equation}{section}

\setlength{\oddsidemargin}{0.2in}
\setlength{\evensidemargin}{0.2in}
\setlength{\textwidth}{6.1in}

\begin{document}



\title{ Math 3210-4,  HW III \\ due Wednesday Oct 15}
\maketitle
\vskip 10pt 
\noindent 
1)  
Let $x_n$ be a sequence of positive real numbers such that $\lim_n x_n =x>0$. Prove that 
\begin{itemize} 
\item 
$\lim_n x_n^2= x^2$.
\item 
$\lim_n \sqrt{x_n}= \sqrt{x}$. 
\end{itemize} 


\vskip 10pt 
\noindent 
2)
Define a sequence by $s_1=1$ and $s_{n+1} = \sqrt{2+\sqrt{s_n}}$. Prove that $s_n <2$ for all $n$, and that $s_n$ is an increasing sequence. 
Hint: prove that $s_{n-1} < s_n$ implies that $s_n < s_{n+1}$. Find the limit. 

\vskip 10pt 
\noindent 
3) Let $X=\mathbb Z$ and $d(x,x)=0$ or $d(x,y)=\frac{1}{2^n}$, if $x\neq y$, where $2^n$ is the largest power of $2$ dividing $x-y$.
Prove that the following two series are Cauchy. One of them is convergent (find its sum) while the other not (explain). 
\begin{itemize} 
\item $\sum_{n=0}^{\infty} 2^n$ 
\item $\sum_{n=0}^{\infty} (-2)^n$ 
\end{itemize} 

\vskip 10pt 
\noindent 
4) Let $c_n$ be a sequence of positive numbers. Prove that 
\[ 
\mathrm{liminf}_n \frac{c_{n+1}}{c_n} \leq \mathrm{liminf}_n \sqrt[n]{c_n}. 
\]  




\vskip 10pt 
\noindent 
5) Let $Y$ be a non-empty set. A function  $d: Y \times Y\rightarrow [0, \infty)$ is called a quasi-distance if 
\begin{itemize} 
\item $d(x,x)=0$  for all $x\in Y$. 
\item $d(x,y)=d(y,x)$  for all $x,y\in Y$. 
\item $d(x,z) \leq d(x,y) + d(y,z)$ for all $x,y,z\in Y$. 
\end{itemize}  
In words, $d$ is almost a distance function, however, $d(x,y)=0$ is allowed for different $x$ and $y$. We say that $x$ and $y$ are 
equivalent if $d(x,y)=0$. Prove that this is an equivalence relation. Prove that, if $x$ is equivalent to $y$ then $d(x,z)=d(y,z)$ for all $z\in Y$. 


\vskip 10pt 
\noindent 
6) Let $(X,d)$ be a metric space.  Let $x=\{x_n\}$ and $y=\{y_n\}$ be two Cauchy sequences. 
Prove that the sequence of distances $d(x_n,y_n)$ is a Cauchy sequence of real numbers. 
%(Whose limit is called a "distance'' between the two Cauchy sequences.) 



\vskip 10pt 
\noindent 
7) Let $(X,d)$ be a metric space.  Let $Y$ be the set of all Cauchy sequence. 
For any two Cauchy sequences  $x=\{x_n\}$ and $y=\{y_n\}$ let 
\[ 
d(x,y)= \lim_n d(x_n,y_n ). 
\] 
Prove that $d$ is a quasi-distance on $Y$. (So when we form equivalence classes as in 5), we get a new metric space, called the completion of $X$.) 
%(In other words, their distance is 0.) 




\end{document} 

\vskip 10pt 
\noindent 
6) Let $(X,d)$ be a metric space.  Two Cauchy sequences $x=\{x_n\}$ and $y=\{y_n\}$ are equivalent if for every $\epsilon >0$, there exists $n$ such that 
$d(x_m,y_m ) < \epsilon$ for all $m\geq n$. Prove that this is an equivalence relation. 
%(In other words, their distance is 0.) 

