Consider a homogeneous and irreducible recurrent Markov chain $\{X_n\}_{n\in\mathbb N}$ defined on a countable state space $S=\mathbb{Z}$. Let $\tau:=\text{inf}\{n\in\mathbb N\,: X_n\ge1\}$ where $\tau$ is defined to be $\infty$, when doesn't exist any $n$ such that $X_n\ge1$. Show that $\mathbf{P}(\tau<\infty)=1.$
Here are a few of my thoughts:
For any $k\in\mathbb{Z}$, define $$\tau_{k}:=\text{inf}\{n\in\mathbb N\,: X_n(\omega)\ge 1,X_{0}(\omega)=k\},$$ then $$\left\{\tau=\infty\right\}=\biguplus_{k\le 0,k\in\mathbb{Z}}\left\{\tau_{k}=\infty \right\}.$$ Suppose that $\mathbf{P}(\tau=\infty)>0,$ there definitely exists $k_{_{0}}\in \left\{0,-1,...,-n,... \right\},$ such that $\mathbf{P}(\tau_{k_{_{0}}}=\infty)>0.$
In an irreducible Markov chain, all states are either transient or recurrent in unison.When all states are recurrent, they are all mutually communicating. Hence,for a fixed $t\in \{1,2,...,n,...\}$,there exists a positive integer $n,$ such that $$\mathbf{P}^{(n)}_{t,k_{_{0}}}:=\mathbf{P}(X_{n}=k_{_{0}}\mid X_{0}=t)>0\Rightarrow$$$$ \exists\mathbf{P}(X_n=k_{_{0}},X_{n-1}=i_{n-1},...,X_{1}=i_{1}\mid X_{0}=t)>0.$$ If there exists $j\in\left\{1,...,n-1\right\}$, such that $X_j=t,$ let $$j_{0}:=\max\left\{j:X_{j}=t,j\in\left\{1,...,n-1\right\}\right\},$$ then $$\qquad\qquad\quad\mathbf{P}(X_n=k_{_{0}},X_{n-1}=i_{n-1},...,X_{j_{_{0}}+1}=i_{j_{_{0}}+1} \mid X_{j_{_{0}}}=t)>0.\qquad\qquad\quad(1)$$
Since $\mathbf{P}(\tau_{k_{_{0}}}=\infty)>0,$ $$\qquad\qquad\qquad\qquad\quad\mathbf{P}(\forall r \in\mathbb{N}_{+},X_{r+n}\notin \{1,2,...,n,...\}\mid X_{n}=k_{_{0}})>0.\quad\qquad\quad(2)$$
$$\begin{align} &(1)\cdot(2)=\\ &\mathbf{P}(\forall r \in\mathbb{N}_{+},X_{r+n}\notin \{1,2,...,n,...\},X_n=k_{_{0}},X_{n-1}=i_{n-1},...,X_{j_{_{0}}+1}=i_{j_{_{0}}+1}\mid X_{j_{_{0}}}=t)>0\\ &\Rightarrow 1-\mathbf{P}(\text{inf}\{n\in\mathbb N_{+}\,: X_n=t\}<\infty\mid X_{0}=t)>0. \end{align}$$ This conflicts with recurrent. I'm not confident about the accuracy of my thoughts, your correction would be appreciate.