This is an exercise from Durret's "Probability: Theory and Examples".
Suppose $p$ is a irreducible and positive recurrent Markov chain. Then $E_x[T_y]<\infty$ for all $x,y$.
I had the following intuition but there is a piece missing:
Since $p$ is positive recurrent, we know that $E_x[T_x]<\infty$ for all x.
Due to $p$ being irreducible there is a $\hat{m}:P_i(X_\hat{m}=j)>0$. Now we can choose $$m=inf \{n \in \mathbb{N} : P_i(X_n=j)>0\}$$ and pick $x_1,\dots,x_{m-1}\neq x$ such that $p(x_1,x_2)\cdot p(x_2,x_3)\cdot \dots \cdot p(x_{m-1},y) >0$.
This is where my problem starts. It seems to be possible to bound $E_x[T_y]$ in terms of $E_x[T_x]$ using these transition probabilities but I can't really see how. I appreciate any suggestions. Thanks!