Let ${x_n}$ be a discrete time Markov Chain with transition probability matrix $P$ and $p_{ii}$<1 for all $i \in S$ . Let ${y_n}$ be a MC with transition matrix $P'$ such that $p'_{i,i}=0$ and $p'_{i,j}=p_{i,j}/(1-p_{i,i})$. Show that $x_n$ is recurrent iff $y_n$ is recurrent.
It would be greatly appreciated if someone could guide me with this proof. My approach was to show $p^{(n)}_{i,j} \leq p^{'(n)}_{i,j}$ for $i \neq j$ so i could sum over both sides but didn't have much luck.