Q) Suppose $S$ is irreducible, $\phi\geq 0$ with $E_x\phi(X_1)\leq \phi(x)$ for $x\notin F$, a finite set and $\phi(x)\rightarrow \infty$ as $x\rightarrow \infty$ i.e., for $\{x:\phi(x)\leq M\}$ is finite for any $M<\infty$, then the chain is recurrent.
If we replace $``\phi(x)\rightarrow \infty"$ by $``\phi(x)\rightarrow 0"$ and assuming $\phi(x)>0$ for $x\in F$, conclude that the chain is transient.
Attempt: Following proof of Theorem $5.3.8$ in Durrett,
$$\text{Let } \tau = \text{inf}\{n>0:X_n \in F\}. \phi(X_{n\wedge\tau}) \text{ is a super martingale.}$$
$$\text{Let } T_M = \text{inf}\{n>0:X_n \in F\text{ or } \phi(X_n)<M\}$$ Since $\phi(x)\rightarrow 0$ as $x\rightarrow \infty$, $\{x:\phi(x)\geq M\}$ is finite and since the chain is irreducible, $T_M<\infty$ a.s. Thus,
$$\phi(x)\geq E_x(\phi(X_{T_M})) = \phi(X_{T_M})P_x(T_M<\tau).1_{\{\phi(X_{T_M})<M\}}+\phi(X_{T_M})P_x(\tau\leq T_M).1_{\{\phi(X_{T_M})\in F\}}$$
$$P_x(\tau\leq T_M) \leq P_x(\tau<\infty)$$
But I'm not sure how to show $P_x(\tau<\infty)<1$ given that $\phi(x)>0$ for $x\in F$? Thanks.