\documentclass{amsart}


  \theoremstyle{plain}  
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
% -------------------------------------------------------------
\begin{document}
\section{Introduction}\label{sec:introduction}

We assume 

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

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

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

\end{document}

