Let $(X_n)_{n\in\mathbb{N}_0}$ be a Markov chain with state space $E$ and transition matrix $(p_{ij})_{i,j\in E}$. Let $C\subseteq E$ be a closed communicating class. Show that $$ (p_{ij})_{i,j\in C} $$ is a stochastic matrix.
First of all I have to say what we mean by "closed communicating class" I think.
For distince $i,j\in E$ we write $i\rightarrow j$ and say that $i$ leads to $j$ if $\mathbb{P}(\exists~n\in\mathbb{N}_0: X_n=j|X_0=i)>0$. For $i,j\in E$ with $i\rightarrow j$ and $j\rightarrow i$ we write $i\leftrightarrow j$ and say that $i$ and $j$ communicate. It can be shown that $\leftrightarrow$ is an equivalence relation. A communicating class is an equivalence class with repsect to $\leftrightarrow$.
Now to my proof:
Let $i\in C$.
Then $i\not\rightarrow j$ for all $j\in E\setminus C$, i.e. $\mathbb{P}(\exists n\in\mathbb{N}_0: X_n=j | X_0=i)=0$, i.e. $p_{ij}^{(n)}=0$ for all $n\in\mathbb{N}_{0}$, especially $p_{ij}^{(1)}=p_{ij}=0$. Furthermore, because $(p_{ij})_{i,j\in E}$ is a stochastic matrix, it is $\sum_{j\in E}p_{ij}=1$. It follows $$ \sum_{j\in C}p_{ij}=\underbrace{\sum_{j\in E}p_{ij}}_{=1}-\underbrace{\sum_{j\in E\setminus C}p_{ij}}_{=0}=1. $$
Is my proof allright?
With greetings