%% This is a source file (a .tex file).   Running LaTeX on a .tex file
%% will produce a pdf file.

%% If you use a different template, make sure that you still have 12-point font, double-spacing,
%% and margins that are as large as the ones this template produces (see comment below
%% about margins).


%% In a .tex file,  any line beginning with a ``%'' sign is invisible to LaTeX. The same
%% goes for all material after a ``%'' on a line that starts with
%% something else.  This allows the writer to insert comment-lines into
%% the file, intended for him/herself or for anyone reading the source
%% file (the .tex file).  I'm using double-%'s in these first few paragraphs
%% to make the comments stand out, but to LaTeX, there's no difference among
%% ``%'', ``%%'', and ``%%%%%%%%%''. 
%% (But note: a line continues until a line-ending command or character, such
%% as <ENTER>.  What may *look* to you like two or more lines in your .tex file may just be one long line, like this one, wrapping each time the width of your text-editor window is reached.  You don't need to insert extra %'s to make that whole long line a comment-line.


\documentclass[12pt]{article}  %% Do not use a font smaller than 12-point for anything in this class.

\usepackage{amsmath,amssymb}

\renewcommand{\baselinestretch}{2.0} % This causes double-spacing in your output, which makes it
                                                             % easier for your professor to make corrections.
                                                            

\setlength{\oddsidemargin}{.75 in} % For your margins to be wide enough for grading, 
                                                   %, the left-side, right-side, top, and bottom margins should
                                                   %  EACH be at least 1.75" (if you were to print out
                                                   %  your pages full-size, on ordinary 8.5" x 11" paper).
                                                   % The \oddsidemargin,\textwidth 
                                                   % \textheight, and  \topmargin settings in this template
                                                   % achieve this.  Do not use settings that make your margins
                                                   % any smaller.   (If you were writing a research paper,
                                                   % you'd use smaller margins, but that's not what this
                                                   % template is for).  

\setlength{\textwidth}{5 in}
\setlength{\textheight}{7.3in}
\setlength{\topmargin}{0.2 in}
\setlength{\parindent}{2em}


%\pagestyle{empty}   %  Keep this commented-out if you want page-numbers.  These will appear at the                           
                                 % bottom of each page.  Un-commenting this \pagestyle command (i.e. deleting
                                 % the "%" at the beginning) will make the page-numbers disappear, and effectively
                                 % make the bottom margin larger.  If you un-comment this line, I recommend using
                                 % \setlength{\textheight}{7.5in} and \setlength{\topmargin}{0.3 in} instead of the
                                 % \textheight and \topmargin values above.


%%%%%%%% some Groisser general-math commands 

\newcommand{\bfc}{{\bf C}}
\newcommand{\N}{{\bf N}}
\newcommand{\bfr}{{\bf R}}
\newcommand{\bfz}{{\bf Z}}
\newcommand{\im}{{\rm im}}
\newcommand{\id}{{\rm id.}}
\newcommand{\image}{{\rm image}}
\newcommand{\Implies}{\Longrightarrow}
\newcommand{\Intersect}{\bigcap}
\newcommand{\iso}{\cong}
\newcommand{\lb}{\langle} % good for inner products
\newcommand{\rb}{\rangle} % good for inner products
\newcommand{\mydot}{{\mbox{$\cdot$}}}  % Good for dots in superscripts



\newcommand{\be}{\begin{equation}} %% starts a numbered equation
\newcommand{\ee}{\end{equation}}
\newcommand{\ben}{\begin{displaymath}} %% Does same thing as \[
   %% (starts an un-numbered, displayed equation)
\newcommand{\een}{\end{displaymath}} %% Does same thing as \]

\newcommand{\pf}{\noi{\bf Proof}: }
\newcommand{\qedns}{\mbox{\hspace{1ex}}
\mbox{\hfill\vrule height10pt width10pt} \\ \vspace{0.5 ex}}
   %% End-of-proof symbol; gives a filled-in box.

%%%%%%%% Groisser Greek-letter commands (just shortens what you have
%%%%%%%% to type).  Wherever you see ``renewcommand'', it
%%%%%%%% means that LaTeX already has a built-in command with this name;
%%%%%%%% you're overriding and replacing that built-in command.

\renewcommand{\a}{\alpha}
\renewcommand{\b}{\beta}
\renewcommand{\d}{\delta}
\newcommand{\e}{\epsilon}
\newcommand{\vare}{\varepsilon}
\newcommand{\g}{\gamma}
\renewcommand{\i}{\iota}
\renewcommand{\k}{\kappa}
\renewcommand{\l}{\lambda}
\renewcommand{\th}{\theta}
\newcommand{\w}{\omega}
\newcommand{\z}{\zeta}

%%%%%%%% some Groisser environments

\newtheorem{theorem}{Theorem}%[section]
\newtheorem{thm}[theorem]{Theorem}
\newtheorem{prop}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{cor}[theorem]{Corollary}
\newtheorem{defn}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}


%%%%%%%% Groisser paragraph-formatting commmands

\newcommand{\noi}{\noindent}
\renewcommand{\ss}{\vspace{.1in}}
\newcommand{\bs}{\vspace{.2in}}
\newcommand{\ms}{\vspace{.15in}}
\newcommand{\bsn}{\vspace{.2in}\noindent}
\newcommand{\msn}{\vspace{.15in}\noindent}
\newcommand{\ssn}{\vspace{.1in}\noindent}

\newcommand{\indenum}{\mbox{\hspace{3ex}}} % This is for indenting
                                % paragraphs in the ``enumerate''
                                % and ``itemize'' environments.



\begin{document}

\title{MAS 4105  Homework 5 [or whatever number]}
\author{Your name here}

\maketitle

% Any line beginning with a ``%'' sign is invisible to LaTeX. The same
% goes for all material after a ``%'' on a line that starts with
% something else.  This allows the writer to insert comment-lines into
% the file, intended for him/herself or for anyone reading the source
% file (the .tex file).  I used double-%'s below to make the
% comments stand out, but to LaTeX, there's no difference between among
% ``%'', ``%%'', and ``%%%%%%%%%''. 

\newpage % optional

\noi {\em Sample start of book problem:}

\ss
\noi \underline{FIS x.y.z  [or x.y/ z.]}  %[Friedberg, Issel, and Spence chapter x, section y, exercise z] 

Let $V$ be  a vector space   \dots


[Optionally, you may write out the problem before your answer. Having
the problem written down on your handed-in work will make it easier
for you to use your homework as a study-aid later.]

\bsn
{\em Sample start of non-book problem:}

\ss
\noi\underline{B1}. (a) \dots

\bsn {\em Sample definition:}

\begin{defn}\rm
  %% Without the \rm, most of the text would come out italicized. In a
  %% definition, the only words/phrases that should be italicized are
  %% those that are being defined. In a theorem (or proposition,
  %% lemma, etc.), the basic text should be italicized, and word(s)
  %% you want emphasized should be in another font.  The {\em ...}
  %% command takes care of the latter automatically (see below); it
  %% toggles between italic font \it and Roman font \rm. 
Let $X$ and $Y$ be sets.  The {\em union} of $X$ and $Y$, denoted $X\cup Y$,
is the set $\{z: z\in X\ \mbox{or}\ z\in Y\}.$
%% The \mbox{} command allows you to insert ordinary words, with spaces between
%% them (if there's more than one word inside the {}), in a math environment.
%% The command "\ [space]" that you see before and after the \mbox{or} above
%% produces space between comes before/after the \mbox{} and the first/last
%% word in the \mbox{}.
\end{defn}

\bsn {\em Sample proposition {\small (written for an Advanced Calculus class; don't 
worry about the content)} , with example of emphasized word, and
  sample proof}:



\begin{prop} For each $n\in\N$, let $f_n:[0,1]\to \bfr$ be the
  function defined by $f_n(x)=x^2+\frac{x}{n}$.  Prove that the
  sequence  $(f_n)_{n=1}^\infty$ converges {\em uniformly}.

\end{prop}

\pf Define $f:[0,1]\to\bfr$ by $f(x)=x^2$. Let $\e>0$ be given. Let
$N$ be any integer greater than $\frac{1}{\e}$.  Then for each $x\in
    [0,1]$ and any $n\geq N$ we have

\begin{align*} %% The ``align'' environment aligns material that's
               %% spread across several lines (with a double-backslash
               %% ending each line), according to where the ampersands
               %% are. The ``split'' environment also does this; I'm
               %% not sure of the difference.  The star in ``align*''
               %% has to be put either in both the \begin and the \end
               %% command, or in neither. Without the stars, you'll
               %% get an equation number on each line.  However, you
               %% can prevent LaTeX from putting an equation number on
               %% any given line by inserting the command
               %% ``\nonumber'' anywhere on that line. The same
               %% general rules for stars and \nonumber apply to any
               %% equation-like environment.
|f_n(x)-f(x)|\  =& \ \frac{x}{n}\\
               %% A backslash followed by a space inserts extra
               %% horizontal space.  Putting more than one space after
               %% the backslash will not affect
               %% the amount of space inserted. 
\leq & \, \frac{1}{n}\\
               %% backslash followed by a comma inserts extra space,
               %% but less than backslash followed by a space.  Spaces
               %% after ``\,'' have no effect on the output.
\leq &  \frac{1}{N}\\
               %% Comment-lines, like this one, are invisible to
               %% LaTeX, but a completely blank line in an
               %% equation-like environment will make LaTeX complain.
               %% Try replacing this comment with blank space--no %
               %% signs--and see what happens.
\leq & \e.
               %% As you can see, with the ``align'' environment, the
               %% output can look cramped if you don't insert extra
               %% space. There is an older alternative, the
               %% ``eqnarray'' environment, that does not have this
               %% defect, but has others. 
\end{align*}

\noi Hence $(f_n)$ converges uniformly (to $f$). \qedns


Here is an illustration of the ``eqnarray'' environment, with a \linebreak
``nonumber'' command on the top line:


\begin{eqnarray}
\nonumber  \z(2) &=& \sum_{n=1}^\infty \frac{1}{n^2}\\
  &=& \frac{\pi^2}{6}\ .
\end{eqnarray}

Some other useful LaTeX commands not illustrated above:

\begin{align*}
& \|x\|  \ \ \ \mbox{(for norms; looks better than $||x||$\ )}\\
 %% To see what  "  \ \ \  "   does, try removing it.  Ditto for  \mbox{ ].   .
& \dots\\
& A\setminus B \\
& {\bf W}\\
& {\mathcal W}\\
& {\tt W}\\
& {\sf W}
\end{align*}


\bigskip
There is a website ``detexify'' that provides a convenient way to
search for LaTeX commands that produce math symbols.  You can also
search using Google, etc., if you know the name of the symbol you're
looking for.


\vspace{0.7in} Above (in the source file) are two examples of
inserting vertical space (using the ``bigskip'' command and the
``vspace'' command). In the ``vspace'' command I used, ``in'' stands
for ``inches''. You can insert space between the number and ``in'';
LaTeX doesn't care. The actual amount of space inserted may not be
what you told LaTeX to insert; for that, you'd use ``truein'' instead
of ``in''.  But usually, you just play around with different-size
spaces till you find one that looks the way you want, without worrying
whether it's exactly the number of inches you told LaTeX to insert.

\newpage

This is a page of text with the margins and spacing produced by this
template file (an example of acceptable margins and spacing for
something you're handing in for grading).  If you're handwriting 
your homework, you may wish to print
out this page and keep it next to you when you're writing.

The quick brown fox jumped over the lazy dog. The quick brown fox
jumped over the lazy dog. The quick brown fox jumped over the lazy
dog. The quick brown fox jumped over the lazy dog. The quick brown fox
jumped over the lazy dog. The quick brown fox jumped over the lazy
dog. The quick brown fox jumped over the lazy dog. 

All work and no play makes Jack a dull boy. All work and no play makes
Jack a dull boy. All work and no play makes Jack a dull boy. All work
and no play makes Jack a dull boy. All work and no play makes Jack a
dull boy. All work and no play makes Jack a dull boy. 

Where did I put that axe? Where did I put that axe? Where did I put
that axe? Where did I put that axe? Where did I put that axe? Where
did I put that axe? Where did I put that axe? Where did I put that
axe? Where did I put that axe? Where did I put that axe? Where did I
put that axe?


\end{document}


