Two unclear points in the proof that every recurrent class is closed I'm learning a theorem about recurrence and transience in Markov chain:



Could you please explain how it follows that


*

*We have $\mathbb{P}_{i}\left(\left\{X_{m}=j\right\} \cap\left\{X_{n}=i \text { for infinitely many } n\right\}\right)=0$.

*$\mathbb{P}_{i}\left(\left\{X_{m}=j\right\} \cap\left\{X_{n}=i \text { for infinitely many } n\right\}\right)=0$ implies $\mathbb{P}_{i}\left(X_{n}=i \text { for infinitely many } n\right)<1$.
Thank you so much!
 A: The first equation follows from the fact that $\ j\not\in C\ $, and therefore does not communicate with $\ i\ $. So if $\ n>m\ $, then
\begin{align}
\mathbb{P}_i\left(\left\{X_n=i\right\}\cap\left\{X_m=j\right\}\right)&=\mathbb{P}_i\left(\left\{X_n=i\right\}\left|\,X_m=j\right.\ \right)\mathbb{P}_i\left(\left\{X_m=j\right\}\right)\\
&=0\ .
\end{align}
That is once the chain reaches the state $\ j\ $ it can never again return to $\ i\ $, let alone do so infinitely often.
The second equation follows from the fact that,
$$
\left\{X_n=i\ \text{for infinitely many } n\right\}=\\
\left(\left\{X_m=j\right\}\cap\left\{X_n=i\ \text{for infinitely many }n\right\}\right)\cup\left(\left\{X_m\ne j\right\}\cap\left\{X_n=i\ \text{for infinitely many }n\right\}\right)\ ,
$$
and therefore
\begin{align}
&\mathbb{P}_i\left(\left\{X_n=i\ \text{for infinitely many } n\right\}\right) =\\&\mathbb{P}_i\left(\left\{X_m\ne j\right\}\cap\left\{X_n=i\ \text{for infinitely many }n\right\}\right)\\
&\le \mathbb{P}_i\left(\left\{X_m\ne j\right\}\right)\\
&= 1-\mathbb{P}_i\left(\left\{X_m= j\right\}\right)< 1\ .
\end{align}
