\documentclass{amsart}

\theoremstyle{plain}  
\newtheorem{thm}{\protect\theoremname}[section]
%\newtheorem{thm}{\protect\theoremname}% => works
\newtheorem{cor}[thm]{\protect\corollaryname}

\begin{document}
\section{Introduction}

We assume 

\begin{thm}
For every $\epsilon>0$, there is a constant $\delta>0$ so that if
\end{thm}

\begin{cor}
For every $\epsilon>0$, there is a constant $\delta>0$ so that if
\end{cor}

\begin{thm}
There exists a positive constant $C_2$ so that
\end{thm}

\end{document}

