\documentclass[12pt, twocolumn]{article}
\usepackage{amsmath}

\newtheorem{theorem}{Theorem}
\newtheorem{observation}[theorem]{Observation}

\begin{document}

This is a much longer document with sections and subsections.

\section{The first section}

Blah blah blah blah blah blah blah blah blah blah blah blah
antidisestablishmentarianistic blah blah blah blah blah blah blah.  Blah
blah blah blah blah blah.  Blah blah blah blah blah $
\sum_{i=\frac{1}{\beta}}^{\infty} \log(\log(\cdots(\log(i))\cdots)) $ blah
blah blah blah blah blah.

\subsection{Here is a subsection}

Here is an unordered list 
\begin{itemize}
\item
{\it{italics}} 
\item
{\underline{underlined}} 
\item
{\tt{typewriter}} 
\end{itemize}

\noindent Here is an ordered list.
\begin{enumerate}
\item
{\tiny{tiny}}.
\item
normal size.
\item
{\Large{Large}}.
\end{enumerate}

\subsection{Another subsection}

Lets put some observations and theorems in here.

\begin{observation}
Most cats have fur.
\end{observation}

\begin{theorem}
If an animal lives underwater it is not a cat.
\end{theorem}

And some math in display mode.
$$
\sum_{i=\frac{1}{\beta}}^{\infty}
\underset{\binom{n}{2}
\mbox{
times}}{\underbrace{\log(\log(\cdots(\log(i))\cdots))}}
$$
\noindent Blah blah blah blah blah blah.  Blah blah blah blah blah.  Blah
blah blah blah blah blah.

\subsubsection{A subsubsection}

Blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah
blah blah blah blah.  Blah blah blah blah blah blah.  Blah blah blah blah
blah.  Blah blah blah blah blah blah.

\subsubsection{Another subsubsection}

Blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah
blah blah blah blah.  Blah blah blah blah blah blah.  Blah blah blah blah
blah.  Blah blah blah blah blah blah.

\section{Another section}

Blah blah blah blah blah blah blah blah blah blah blah blah blah blah blah
blah blah blah blah.  Blah blah blah blah blah blah.  Blah blah blah blah
blah blah blah blah blah blah blah blah blah blah blah blah blah blah
blah.  Blah blah blah blah blah blah.

\end{document}