Q) Let $X_n$ be a random walk on $\{0,1,\cdots\}$ s.t. $p(i,i+1)=p_i, p(i,i-1)=1-p_i=q_i$. $p(0,0)=1-p_0$. Show that $X_n$ is positive recurrent if and only if:
$$\sum_i \prod_{0<j\leq i} \frac{p(j-1)}{q(j)} < \infty$$
Durrett has a claim saying that the chain is recurrent if and only if:
$$\sum_i \prod_{0<j\leq i} \frac{q(j)}{p(j)} = \infty$$
the proof of which I understand but may I know how to go from recurrence to showing positive recurrence?