I am reading the paper "exact sampling and approximate counting technique". And I got confused on a step in page $5$. section $5.1$, proof the Theorem $3$ part 1, which involved using of Wald's lemma.
Theorem $3$, part 1 state as follow
Let $N_t$ be a random walk on $\{0, \dots n \} $ with $n$ a absorbing state and $0$ a reflection state. Suppose that $N_0=0$, $|N_{t+1}- N_{t}| \leq s$. $ Pr(N_{t+1} \neq N_t) > 0$, and $w_i$ is the expected number of times that $N_t=i$
if $\mathbb{E}(N_{t+1}- N_t) \geq 0$ then
$$w_i \leq \frac{2s(n-i)}{Pr(N_{t+1} \neq N_t|N_t =i)}$$
In the proof, he claims that: let $v_i$ denote the expected number of times $N_t =i$ and and $N_{t-1}\neq i$. Then by Wald's lemma, we have that
$$w_i \leq \frac{v_i}{Pr(N_{t+1} \neq N_t|N_t =i)}$$
How Wald's lemma can be apply here?