Consider a Markov chain over the state space $\mathbb{Z}$ with the following transition probabilities: $$p(n+1|n)=(1-\epsilon)x_n$$ $$p(n|n)=1-(1-\epsilon)x_n-\epsilon(1-x_n)$$ $$p(\max\{n-1,0\}|n)=\epsilon(1-x_n)$$ where $1>\epsilon>0$, $1>a>x_n>b>0$ for some $a,b$.
So the chain at each state $n$ can go up by 1, down by 1 or remain unchanged, except the lower bound for $n=0$.
Assume $f:\mathbb{Z}\rightarrow\mathbb{R}$ is a bounded function. For every $\lambda\in(0,1)$, define $$g(r,\lambda)=E\Big[(1-\lambda)\sum_{t=0}^\infty \lambda^tf(X_t)|X_0=r\Big].$$
It is very intuitive to me that when $\lambda$ is very close to 1, the value $g(r,\lambda)$ should be very close to $g(r+1,\lambda)$. I'm wondering if this is true or not. If it is true, is there any formal statement about this? In particular can we find $M>0$ independent of $\lambda$ such that $$\sup_r |g(r,\lambda)-g(r+1,\lambda)|<(1-\lambda)M$$