To recall the basic set up, if we have a (time homogeneous) Markov Chain $X_n$, we say that states $i$ and $j$ communicate if there exist $n$ and $m$ such that $P_i(X_n=j)>0$ and $P_j(X_m=i)>0$. (Recall that $P_i$ is just the conditional probability measure on the event $X_0=i$; i.e., $P_i=P(\cdot\vert X_0=i)$.) Communication is an equivalence relation and partitions the state space into disjoint sets. We will call the elements of the partition classes.
Now, in a book on stochastic processes, they say that a class is transient if the M.C. starts in the class then eventually leaves and never returns with probability 1. Mathematically, a class $T$ is transient if $$P\left(\bigcup_{N\ge 1}\bigcap_{n\ge N} \{X_n\in T^c\}\,\vert\, X_0\in T\right)=1.$$ The states in the transient class are called transient states. They then define a recurrent class as a class that is not transient; i.e., $R$ is recurrent if $$P\left(\bigcup_{N\ge 1}\bigcap_{n\ge N} \{X_n\in R^c\}\,\vert\, X_0\in R\right)<1.$$ Again, the states of $R$ are called recurrent states.
I recognize that one usually defines the terms recurrent and transient in a slightly different way; i.e., usually one starts by defining recurrent and transient states, opposed to the classes. And, furthermore, the definition usually is stated with the probability that a state returns to itself in finite time, opposed to whether the chain stays in the class. However, I would like to work with the definition I provided. (Unless of course something is wrong, in which case I would like a correct definition in terms of classes.)
I would like to prove some familiar things about recurrent classes. For example, I would like to prove the following:
(i) If $R$ is recurrent then $P\left(\bigcap_{n\ge 1}\{X_n\in R\}\,\vert\, X_0\in R\right)=1$, i.e. if the chain starts in $R$ then it never leaves with probability $1$.
(ii) If $R$ is recurrent then $P\left(\bigcap_{N\ge 1}\bigcup_{n\ge N}\{X_n=i\}\,\vert\, X_0=i\right)=1$ for all $i\in R$, i.e. the chain will return to $i$ infinitely often if it starts at $i$.
I wasn't having too much luck myself trying to rigorously prove these using the definitions of transient and recurrent classes that I provided. Would appreciate help to proving these.