Two unclear points in the proof that every recurrent class is closed

Question

I'm learning a theorem about recurrence and transience in Markov chain:

Could you please explain how it follows that

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

2. $\mathbb{P}_{i}\left(\left\{X_{m}=j\right\} \cap\left\{X_{n}=i \text { for infinitely many } n\right\}\right)=0$ implies $\mathbb{P}_{i}\left(X_{n}=i \text { for infinitely many } n\right)<1$.

Thank you so much!

Answer 1

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}