Let $\{X_n\}$ be an irreducible DTMC on a finite state space $\mathbb{X}$, such that $p(x,x)=0$ for all $x\in \mathbb{X}$. For $y\in \mathbb{X}$, let $T_y=\inf\{k\geq 0: ~X_k=y\}.$ True or false: for every $x,~y\in \mathbb{X}$ we have $P(T_y<+\infty~|~X_0=x)=1.$
I believe the answer is yes (based on the assumption that $p(x,x)=0$ for all $x\in \mathbb{X}$). Since $p(x,x)=0$, we have that our chain reaches a state $x_1$ with probability $1$ in one step. At the $n-$th step our chain has reached $x_1,x_2,\ldots,x_n$, succesively, with prob. $1$. I am not that sure though why our chain reaches $y$ at some finite time.
Thanks a lot in advance.