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\documentclass[12pt]{article}

\usepackage{amssymb,amsmath,amsthm}
\usepackage[top=1in, bottom=1in, left=1.25in, right=1.25in]{geometry}
\usepackage{fancyhdr}
\usepackage{enumerate}

% Comment the following line to use TeX's default font of Computer Modern.
\usepackage{times,txfonts}

\newtheoremstyle{homework}% name of the style to be used
  {18pt}% measure of space to leave above the theorem. E.g.: 3pt
  {12pt}% measure of space to leave below the theorem. E.g.: 3pt
  {}% name of font to use in the body of the theorem
  {}% measure of space to indent
  {\bfseries}% name of head font
  {:}% punctuation between head and body
  {2ex}% space after theorem head; " " = normal interword space
  {}% Manually specify head
\theoremstyle{homework} 

% Set up an Exercise environment and a Solution label.
\newtheorem*{exercisecore}{Exercise \@currentlabel}
\newenvironment{exercise}[1]
{\def\@currentlabel{#1}\exercisecore}
{\endexercisecore}

\newcommand{\localhead}[1]{\par\smallskip\noindent\textbf{#1}\nobreak\\}%
\newcommand\solution{\localhead{Solution:}}

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% Stuff for getting the name/document date/title across the header
\makeatletter
\RequirePackage{fancyhdr}
\pagestyle{fancy}
\fancyfoot[C]{\ifnum \value{page} > 1\relax\thepage\fi}
\fancyhead[L]{\ifx\@doclabel\@empty\else\@doclabel\fi}
\fancyhead[C]{\ifx\@docdate\@empty\else\@docdate\fi}
\fancyhead[R]{\ifx\@docauthor\@empty\else\@docauthor\fi}
\headheight 15pt

\def\doclabel#1{\gdef\@doclabel{#1}}
\doclabel{Use {\tt\textbackslash doclabel\{MY LABEL\}}.}
\def\docdate#1{\gdef\@docdate{#1}}
\docdate{Use {\tt\textbackslash docdate\{MY DATE\}}.}
\def\docauthor#1{\gdef\@docauthor{#1}}
\docauthor{Use {\tt\textbackslash docauthor\{MY NAME\}}.}
\makeatother

% Shortcuts for blackboard bold number sets (reals, integers, etc.)
\newcommand{\Reals}{\ensuremath{\mathbb R}}
\newcommand{\Nats}{\ensuremath{\mathbb N}}
\newcommand{\Ints}{\ensuremath{\mathbb Z}}
\newcommand{\Rats}{\ensuremath{\mathbb Q}}
\newcommand{\Cplx}{\ensuremath{\mathbb C}}
%% Some equivalents that some people may prefer.
\let\RR\Reals
\let\NN\Nats
\let\II\Ints
\let\CC\Cplx

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% The following commands set up the material that appears in the header.
\doclabel{Math 401: Homework 6}
\docauthor{Your name goes here!}
\docdate{October 5, 2020}

\begin{document}


\begin{exercise}{2.4.5 (Modified, with hints!)}
Suppose $x_1=2$ and define
\[
x_{n+1} = \frac{1}{2}\left(x_n + \frac{2}{x_n}\right).
\]
\begin{enumerate}
	\item Show that $x_n\ge 0$ for all $n$.
	\item Show that if $a>0$ then $a + \frac{1}{a} \ge 2$.
	Hint: $(a-1)^2 \ge 0$.  [Your proof should highlight the part where you use the hypothesis $a>0$.]
	\item Show that if $b\neq 0$ then 
	$b^2+4/b^2 \ge 4$. Hint: Use the previous item!
	\item Show that $x_n^2 \ge 2$ for all $n$. Hint: Use the previous item!
	\item Show that $x_n \ge x_{n-1}$ for all $n$.  Hint: Use the previous item!
	\item Show that the sequence converges to a limit $L$.
	\item Show that $L\neq 0$. Hint: If $x_n\to 0$ then $x_n^2\to 0$.
	\item Show that $L^2=2$. Hint: $\lim x_{n+1} = \lim x_n$.
\end{enumerate}
\end{exercise}

\begin{exercise}{2.5.5} 
\end{exercise}

\begin{exercise}{2.5.6} 
\end{exercise}

\begin{exercise}{2.5.7} 
\end{exercise}

\begin{exercise}{2.6.2} 
\end{exercise}

\begin{exercise}{2.6.7 (b)} 
\end{exercise}

\end{document}