diff --git a/main/2048isAcyclic.tex b/main/2048isAcyclic.tex
index 94458ce..36e3256 100644
--- a/main/2048isAcyclic.tex
+++ b/main/2048isAcyclic.tex
@@ -1,9 +1,5 @@
-\section{Aacyclicity of the 2048 game}
 
 
-The purpose of this section is to prove the acyclicity of the 2048 game
-and give some discussions.
-
 \subsection{acyclicity of the 2048 game}
 
 The 2048 game consists of a 4$\times$4 grid board, totaling 16 squares.
diff --git a/main/acyclic.tex b/main/acyclic.tex
index e4a2d5c..969d227 100644
--- a/main/acyclic.tex
+++ b/main/acyclic.tex
@@ -1,4 +1,9 @@
-\section{Acyclicity between non-absorbing states}
+\section{Aacyclicity of the 2048 game}
+
+
+The purpose of this section is to prove the acyclicity of the 2048 game
+and give some discussions.
+\subsection{Acyclicity between non-absorbing states}
 \begin{definition}[Acyclicity between non-absorbing states]
 Assume that $N$  exists for any policy $\pi$
  and is independent of initial states.
@@ -8,7 +13,8 @@ Assume that $N$  exists for any policy $\pi$
   \label{definition3}
 \end{definition}
 
-
+It is easy to see that if a Markov chain is ergodic,
+then it is cyclic.
 
 
 \subsection{Boyan chain}
@@ -65,7 +71,7 @@ Bases on Definition \ref{definition3},
 Boyan chain 
 is acyclic between non-absorbing states.
 
-\subsection{A sufficient condition for acyclicity between non-absorbing states}
+%\subsection{A sufficient condition for acyclicity between non-absorbing states}
 By observing Boyan chain, 
 it is easy to provide a sufficient condition for acyclicity between non-absorbing states.
 \begin{theorem}[A sufficient condition for acyclicity between non-absorbing states]