# how to format under the following requirements?

I'm submitting an article to the Journal of Combinatoric. Now I need to follow these two requirements.

(1): In a paper divided into sections, it is desirable to number theorems, lemmas, definitions, corollaries, examples, etc. consecutively using double Arabic numerals. (E.g., Section 3 may start with Definition 3.1 followed by Remark 3.2 and Theorem 3.3).

(2): Mark the end of proof by ∎. The same symbol, placed at the end of a statement, indicates the absence of proof.

The paper I have currently is like the following: (1): The paper is divided into sections. However, the theorems, lemmas, definitions, corollaries, examples, etc, are numbered consecutively, which has nothing to do with the section order what so ever; (2): The end of the proof is marked by a blank ∎, not ∎.

Do you know how to accomplish these two tasks? Many thanks for your time and attention.

The command \qed is defined in amsthm. By default it is a non-filled square, but can be changed by \renewcommand\qedsymbol{\ensuremath{\blacksquare}}. To use \blacksquare include also amssymb.

To number the theoremns etc withing the section you can use \newtheorem{theorem}{Theorem}[section].

An example (if I understand the requirements right):

\documentclass{article}
\usepackage{amsthm}
\usepackage{amssymb}
\renewcommand\qedsymbol{\ensuremath{\blacksquare}}
\newtheorem{theorem}{Theorem}[section]
\usepackage{lipsum}
\begin{document}
%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
\label{Sec:Intro}
\lipsum[1]
\begin{theorem}
\label{Th:Obvious}
Let $a$, $b$ and $c$ be real numbers. If $a\leq b$ and $b\leq c$, then $a\leq c$.
\qed
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%
\section{Complicated Theory}
\label{Sec:Complicated}
\lipsum[2]
\begin{theorem}
\label{Th:Complicated}
Let $a$, $b$ and $c$ be real numbers. If $a\leq b$ and $b\leq c$, then $\sum_{x\in\{a,b,c\}}x\leq 3c$.
\end{theorem}
\begin{proof}
Follows directly from Theorem~\ref{Th:Obvious}.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%
\end{document}