2

Here is a MWE:

  %\documentclass{beamer}
\documentclass{article}
\usepackage{beamerarticle}
\begin{document}

\section{First section}

\begin{frame}
\begin{definition}[Gnats]
Gnats are small
\end{definition}

\begin{theorem}[Gnus]
Gnus are large
 \end{theorem}

 \begin{corollary}
 Animals come in different sizes
 \end{corollary}
\end{frame}


\section{Second section}

\begin{frame}
\begin{fact}[Bees]
Bees do it
\end{fact}
\begin{lemma}[Birds]
Birds also do it
 \end{lemma}
 \begin{theorem}[us]
Let's do it!
  \end{theorem}
\end{frame}
\end{document}

The output just numbers everything together.

2

There's the option for it:

\usepackage[envcountsect]{beamerarticle}

enter image description here

\documentclass{article}
\usepackage[envcountsect]{beamerarticle}

\begin{document}

\section{First section}

\begin{frame}
\begin{definition}[Gnats]
Gnats are small
\end{definition}

\begin{theorem}[Gnus]
Gnus are large
 \end{theorem}

 \begin{corollary}
 Animals come in different sizes
 \end{corollary}
\end{frame}


\section{Second section}

\begin{frame}
\begin{fact}[Bees]
Bees do it
\end{fact}
\begin{lemma}[Birds]
Birds also do it
 \end{lemma}
 \begin{theorem}[us]
Let's do it!
  \end{theorem}
\end{frame}
\end{document}

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