\documentclass[11pt]{article}
\usepackage[top=1.5 in, left=1.5 in, bottom=1.5 in, right=1.5 in]{geometry}
\setlength{\parindent}{0pt}  %no indentation
\setlength{\parskip}{0.1 in} %extra vertical space between paragraphs
\begin{document} 
\title{Latex Example 2}
\author{Karen Mortensen and revised by GF 28dec09}
\maketitle
\textbf{ Theorem.}\textit{ For all real numbers $x$ and $y$, \[|xy|=|x||y|\].}

\textbf{ Proof.} To prove this, first suppose that $x \ge 0$ and $y \ge 0$.  
Then $xy \ge 0$.
By definition of absolute value, $|xy| = xy$, $|x| = x$ and $|y|=y$. 
Therefore $|xy| = |x| |y|$.

Next suppose that $x \ge 0$ and $y < 0$.  Then $xy \le 0$.  By definition,
$|xy| = -(xy)$, $|x| = x$ and $|y| = -y$.  Since $ -(xy) = x(-y)$, we conclude 
that $|xy| = |x| |y|$.

Next suppose that $x < 0$ and $y \ge 0$. The argument from the previous 
paragraph, with the roles of $x$ and $y$ reversed, shows that $|xy| = |x| |y|$.

Finally, suppose that $x < 0$ and $y < 0$.  Then $xy >0$.  By definition,
$|xy| = xy$, $|x| = -x$ and $|y| = -y$.  Since $xy = (-x) (-y)$, we conclude
that $|xy| = |x| |y|$.

Having considered all possible cases for the signs of $x$ and $y$, we have
proved that $|xy| = |x| |y|$.
\hfill q.e.d.

This proof is of importance because it establishes that absolute value 
commutes with multiplication, a commonly used property of absolute value.
The proof illustrates the strategy of dividing a mathematical statement into
several cases and proving each case separately.  Once the statement was broken 
tiny bit of algebra.  This demonstrates the importance of going back to 
definitions when constructing proofs.

\end{document}

% LATEX NOTES
%
% 0. The \usepackage[...]{geometry} line is a most convenient way of handling
% the margins of your document. In a multipage document, experiments with 
% margins can avoid widows and orphans without learning more commands for it. 
%
% 1. A line beginning with % is a ``comment line'' and has no effect on the 
% printed version of the document.
%
% 2. Every document begins with the line  \documentclass[ nn pt]{article} 
% where nn = 9 (default), 10, 11, 12, the typeface sizes in points (1/72 in)
%
%3.  The documentclass "article" has its own ideas how your text should be typset. But you
%can change that by explicitly overriding the defaults. For example the optional lines that
%come before the \begin{document} such 
%\setlength{\parindent}{0pt}
% which causes a new paragraph to be indicated by a blank line rather than by indenting.

% 4. All of the text of the document must be between the lines  \begin{document}  and \end{document}.
%
% 5. The lines \title{Latex Example 1}
%              \author{Karen Mortensen}
%              \maketitle
% create the heading of the printed document.  The date will be put in automatically.
%
% 6. In LaTeX, { } are grouping symbols which do not appear in the printed document.  Use the commands
% \{ and \} instead if you want these curly brackets to appear in the printed document.
%

% 7. \textbf{ Theorem.} will cause the word Theorem. to appear in bold face.  The curly brackets { } are
% needed to tell LaTeX where to begin and end the bold face.
%
% 8. The command \vspace{.25 in} causes LaTeX to leave a quarter inch vertical space. 
%I've put this before and after the  proof to make the document look a little nicer.
% 
% 9. All mathematical symbols must go between dollar signs $ $.  The $ will not be printed, but simply
% indicates where the mathematical symbols begin and end. This should be used even if a variable x
% appears alone in a sentence:  put $x$, since this affects the typeface used.
% If your text editor doesn't check for matching $, it might be wiser to 
% use \( x \) instead of $ x $. It's more modern too.
% Math text between \[ and \] sets it off. 
% Older people might use $$ for both \[ and \].
% 10. The command \ge makes the symbol for ``greater than or equal to''.  The command \le makes the symbol 
% for ``less than or  equal to.  There are many other such commands - see the handout you got in class.
%
% 11. Extra blank spaces have no effect on the printed document, so put in blank spaces as much as you
% like to make the LaTeX file easier to read.  Same for new lines.  A blank line, however, will cause a new
% paragraph to be started.
%12. Common keyboard symbols, like { } \ %  have special meaning for TeX, as you have 
%seen above. If you want to print what they regularly mean, you must "escape" them by
% prefixed \ .  Thus TeXing \\ will produce just a single backslash. 
%
%13. On the other hand, when you escape a keyboard symbol that doesn't already
%have a special meaning, such as [ , the \[ has a new meaning, namely the beginning
% of displayed math formula. Indeed, \bf followed by a space comes to mean bold-face.
% In this way, TeX can be extended to have an arbitrary large vocabulary. 
%
%14. TeX always prints " as end-quotes. They slant to the right.
%Beginning quotes, which slant to the left, are written thus ``. Don't forget!

