I am having trouble reaching the final conclusion of Foster's theorem regarding a sufficient condition for recurrence in discrete time Markov chains.
Theorem: Let $X$ be an irreducible discrete time Markov chain with transition matrix $P$ and state space $S$. If there exists a finite set $F \subset S$ and a non negative function $h:S \to \mathbb{R}$ such that for all $x \notin F$ the following hold:
$h(x) \ge \sum_{y \in S}{P(x,y)h(y)}$
for all $r\in \mathbb{R} $ the set $A_r := \{x\in S | h(x) \le r \} $ is finite
then $X$ is recurrent.
Proof:
Assume $X_0 \notin F$
Define $T_F := \inf\{n \ge 0 |X_n \in F \}$ and define $Y_n := h(X_{\min(n,T_F)})$
Then $Y$ is a non negative supermartingale and hence converges almost surely to a finite limit. By the property regarding the sets $A_r$ we have that $Y_n$ achieves at most a finite number of values.
This can happen in one of two ways. Either $T_F < \infty$ implying that $Y_n := h(X_{\min(n,T_F)})$ is constant for $n \ge T_F$, or $T_F = \infty$ and $ X_n$ achieves only a finite number of values.
I was able to prove that $T_F < \infty $ implies $X$ is recurrent by a pigeonhole principle argument. I understand that i am supposed to show that the second way in which $Y_n$ can converge is impossible, but i am stuck.
Any help on how to approach this is appreciated, Thanks!