Theorems and Proof Environments in LaTeX

\usepackage{amsthm}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}

\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

\begin{theorem}[Convergence]
Let $f_n$ be a Cauchy sequence in a complete normed space. Then $f_n$ converges.
\end{theorem}

\begin{proof}
Apply completeness and standard norm arguments.
\end{proof}