diff --git a/main/2048isNonergodic.tex b/main/2048isNonergodic.tex
index 5017391..47e7179 100644
--- a/main/2048isNonergodic.tex
+++ b/main/2048isNonergodic.tex
@@ -8,6 +8,7 @@
 
 \end{IEEEproof}
 
+%\input{material/2048prove}
 
 
 
diff --git a/main/background.tex b/main/background.tex
index 680badb..7e7e6f1 100644
--- a/main/background.tex
+++ b/main/background.tex
@@ -1,5 +1,21 @@
 \section{Background}
-
+\subsection{2048 game rules}
+The 2048 game consists of a 4$\times$4 grid board, totaling 16 squares.
+ At the beginning of the game,  two squares are randomly filled
+  with tiles of either 2 or 4.
+  Players can make moves in four directions: \textit{up}, \textit{down},
+   \textit{left}, and \textit{right}. 
+   When a player chooses a direction, 
+   all tiles will move in that direction until 
+   they hit the edge or another tile. 
+   If two tiles with the same number are adjacent 
+   in the moving direction, they will merge into 
+   a tile with the sum of the original numbers.
+   Each tile can only participate in one merge operation per move.
+   After each move, a new tile appears on a random empty square.
+   The new tile is 2 with  probability 0.1, and 4 with probability 0.9.
+The game ends when all squares are filled, and no valid merge operations can be made.   
+\subsection{MDP}
 Consider Markov decision process (MDP)
 $\langle \mathcal{S}$, $\mathcal{A}$, $\mathcal{R}$, $\mathcal{T}$$\rangle$, where 
 $\mathcal{S}=\{1,2,3,\ldots\}$ is a finite state space, $|\mathcal{S}|=n$, $\mathcal{A}$ is an action space,
diff --git a/main/nonergodic.tex b/main/nonergodic.tex
index 42a982f..2a5f786 100644
--- a/main/nonergodic.tex
+++ b/main/nonergodic.tex
@@ -92,10 +92,9 @@ it is easy to provide a sufficient condition for non-ergodicity between non-abso
 \begin{theorem}[A sufficient condition for non-ergodicity between non-absorbing states]
 Given a Markov chain with absorbing states, 
 suppose the size of the non-absorbing states $|S\setminus\{\text{T}\}|\geq 2$.
-If the transition probabilities $Q$ between non-absorbing states satifies,
-$\forall i,j \in S\setminus\{\text{T}\}$,
+If the transition matrix $Q$ between non-absorbing states satifies,
 \begin{equation}
-Q_{i,j}=\begin{cases}
+\forall i,j \in S\setminus\{\text{T}\}, Q_{i,j}=\begin{cases}
 \geq 0, & \text{if } i\leq j; \\
  0, & \text{otherwise.}  
 \end{cases}
@@ -104,7 +103,7 @@ Q_{i,j}=\begin{cases}
 Then, the Markov chain is non-ergodic between non-absorbing states.
 \end{theorem}
 \begin{IEEEproof}
-Based on the assumption, the $Q$ matrix is an upper triangular matrix.
+The $Q$ matrix (\ref{condition}) is an upper triangular matrix.
 The product of two upper triangular matrices is still an upper triangular matrix.
 Furthermore, the sum of two upper triangular matrices 
 is still an upper triangular matrix.