Assume $\mathcal{S} := \{0, 1, \cdots \}$, $p(0,1)=1$ and $p(n,0)=p(n,n+1)=\frac{1}{2}$ for $n=1,2, \cdots$. Is $0$ recurrent or transient?
So, basically this is an irreducible, closed but infinite Markov chain. Hence, we know that either all states are recurrent or transient.
If I define $\rho_{n,0} := P(T_0 < \infty \; | \; X_0 = n) $ where $T_0 := \inf \{ n \ge 1 : X_n =0 \}$, we will have the following recursive relation
\begin{align} \rho_{n,0} = \frac{1}{2}(\rho_{n+1,0} + 1) \end{align}
so it seems that there is always positive probability for not coming back to 0 (simply keep moving to right). Can we argue that 0 is transient then?
HINT probability to never return to 0 starting at 0 is to always move to the right, i.e. $$\lim_{n \to \infty} (1/2)^n = 0$$