markov chain - transient and recurrent states proof Claim: "All communicating classes, $C$, are either transient or recurrent"
My tactic was to show 'if $i\in C$ is transient and $i\leftrightarrow j$ then $j$ must be transient'. I was wondering is this will suffice to prove the claim? 
Do I also need to show the scenario where if $i\in C$ is recurrent and $i\leftrightarrow j$ and so $j$ is recurrent as well to be complete with the proof?
 A: This is a logic question and has not much to do with Markov Chains.  The key questions are:


*

*Is $i \leftrightarrow j$ a symmetric relation, i.e. Does $i\leftrightarrow j$ iff $j \leftrightarrow i$? 

*Is transient and recurrent complements of each other, i.e. is a state either transient, or recurrent, never both (and never neither)? 
In Markov Chains, of course the answers to the above questions are all Yes.
When answers are Yes, you don't need to prove both directions because one follows from the other.  


*

*You have already proved LEMMA: $i \leftrightarrow j$ and $i$ transient $\implies j$ transient.

*Now given: $i \leftrightarrow j$ and $i$ recurrent.

*Assume for future contradiction that $j$ not recurrent.

*Since "not recurrent" = "transient", therefore $j$ transient.

*Since $\leftrightarrow$ is symmetric, therefore $j \leftrightarrow i$.

*You have $j \leftrightarrow i$ and $j$ transient, so by LEMMA: $i$ transient.  This contradicts the given condition.  Thus the assumption (that $j$ not recurrent) must be false. $\square$
As you can see, you don't need anything from Markov Chain, except the first two questions.
