\documentclass[12pt,draft]{article}
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\usepackage{amsmath} % ams = American Mathenatical Society
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\newcommand{\LaTex}{It's not called ``LaTex''!}
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\newcommand\paren[1]{\left(#1\right)}
\newcommand\seq[1]{\paren{#1}}
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\newcommand\nats{{\mathbb N}}
\newcommand\rats{{\mathbb Q}}
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\newcommand\divides{\mathrel{|}} % mathord, mathbin, mathrel
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\newenvironment{remark}
    {\noindent\(\spadesuit\)~\textbf{Remark.}}
    {\hfill\(\heartsuit\)}
\newtheorem{theorem}             {Theorem}    [section]
\newtheorem{corollary} [theorem] {Corollary} 
\newtheorem{lemma}     [theorem] {Lemma}     
\newtheorem{definition}          {Definition} [section]

\begin{document}

\tableofcontents

\newpage

\section{Introduction}
If you want to see math, you should look at section \secref{important}, which is on page \pageref{math}.
bém
Hello \textbf {there}.
{How \bfseries are you?%
\footnote{I don't really care.}
I am} th\"an\c os.  Wh\^ o are you?  I am Nikos.% hi

\begin{description}
    \item[Tigolo]
        Aquele neg\'ocio
        Aquele neg\'ocio
        Aquele neg\'ocio
        Aquele neg\'ocio
        Aquele neg\'ocio
        Awkd kwa hdakwu hdlkwuhd alkw hdlkwa hdlkwau hdkhuquele neg\'ocio
    \item[Chaves]
        Usamos para grupar.
\end{description}

\begin{multicols} 2
\begin{enumerate}[-] % enumerate, itemize, description
\item[Isso:] First item
\item[Ou isso:] Second item
\item {\rmfamily I am roman (serif).}
\item {\sffamily I am sans serif {\itshape and italics!}.}
    \begin{enumerate}[(i)]
        \item Hi
        \item I am nested
    \end{enumerate}
\item {\ttfamily I am typewritter.}
\item {\mdseries I am metropole digital.}
\item {\bfseries I am boldface.}
\item {\upshape  I am upright.}
\item {\itshape  I am italics.}
\item {\slshape  I am slanted.}
\item {\scshape  I am Small Capitals.}
\end{enumerate}
\end{multicols}

Let's see some differences.
This is \LaTex, not \TeX!

\newpage

\section[How to write math]{How to write math---Not very detailed}
\subsection{General math}
\label{math}
Let \(f\) be a function from a set \(A\) to a set \(B\).
Then,
\begin{equation}
    f(x) = \frac a b.
\label{eq:important}
\end{equation}
Is the se-quence, pages 3--12, He started running and---Fuck, he started
rinning indeed---then he remembred something.
\(\seq{x_{12}, x^{2x^{4a_5}+2}_2, \seq{1,2,3}, \frac{x_3} 9}\)
nice?

\bigskip

\begin{center}
Hi.
\end{center}

\medskip

\begin{center}
\rule{\textwidth}{1mm}
\end{center}

\smallskip

I prefer the set \(\set{0, 1, 2, \dotsc}\).  Look at the sum:
\[
    1 + \frac {13} 2 + \frac 1 4 + \dotsb \neq 2.
\]

\subsection{Nested scope}
{\newcommand\R{\mathrel{R}}
Let \(n \in \nats\), \(x \in \reals\),
\(\seq{u,v}\in\reals^2\) and \(b\in\bools\).
Suppose that \(2 \divides n\) and \(2\ndivides m\).
\[
x \R y.
\]
Let \(f\) be defined by
\begin{align}
    f(0)   &= 0               & g(0)   &= 3\tag{TI}\label{eq:ti}\\
    f(1)   &= 1               & g(1)   &= 5\notag\\
    f(n+2) &= f(n) + f(n+1)   & g(n+2) &= \frac{g(n-1)}{g(n)}\label{eq:rec}.
\end{align}
}

\subsection{More align}
Using \eqref{eq:ti} and \eqref{eq:rec} I obtain the result.

We compute
\begin{align*}
    f(x,y,z)
    &= x^2 + g(x,y)   &&\text{(Def.~of \(f\))}\\
    &= x^2 + x + y    &&\text{(Def.~of \(g\))}\\
    &= x(x + 1) + y  \\
    &= 42             &&\text{(Obvious, by \eqref{eq:ti})}
\end{align*}

I think I will have a nice and delicious dinner with
my friend calledermenter\-mente D.~Knuth tonight.
We saw this on p.~\pageref{eq:important}

\section{Sizes and other nifty things}
{
    \Huge
hello    \huge
hello    \LARGE
hello    \Large
hello    \large
hello    \normalsize
hello    \small
hello    \footnotesize
hello    \scriptsize
hello    \tiny
hello We saw in equation \eqref{eq:important}.
}

\section{Important things}
\label{sec:important}
\begin{remark}
We can easily \emph{emphasize} using the \texttt{emph} command.
Observe that it is smart enough to know the shape of its context.
\end{remark}

\begin{theorem}
    There are infinite sets.
\end{theorem}

\begin{theorem}\label{thm:no-infinite}
    There are no infinite sets.
\end{theorem}

\def\qedsymbol{\textsc{qed}}
\begin{proof}
    name
    This is obvious.
    Let us calculate the limit
    \[
        \lim_{x\to \infty}
        \max_{x\in\reals}
        \mathsf{isFinite}(x)
    \]
\end{proof}



Using Theorem~\ref{thm:no-infinite}, we obtain the following:
\begin{corollary}
    Things are nice.
\end{corollary}

\begin{definition}
    There are no infinite sets.
\end{definition}

\begin{remark}
    This is the way to make remarks.
\end{remark}

Bla bla this is a paragraph.
Bla bla this is a paragraph.
Bla bla this is a paragraph.
Bla bla this is a paragraph.
Bla bla this is a paragraph.
Bla bla this is a paragraph.

\vspace{\stretch 1}

Bla bla this is a paragraph.
Bla bla this is a paragraph.

\vspace{\stretch 3}

Bla bla this is a paragraph.
Bla bla this is a paragraph.
Bla bla this is a paragraph.
Bla bla this is a paragraph.

\begin{remark}
    I like to make remarks.  Let \(f : A \to B\).
\end{remark}

\section{Some useful stuff}
\begin{description}
\item[tikz] create graphics
\item[beamer] create presentations
\item[memoir] a class for (larger) documents
\item[bibtex] for bibliographies
\item[listings] typeset source code
\end{description}

\end{document}
