Problem
Consider a random walk $X$ on $\mathbb{Z}$ defined by $p(1)=p(0)=p(-1)=1 / 3$ (note that $\left.p_{x, y}=p(y-x)\right)$. Define $$ \begin{aligned} T &:=\inf \left\{n \geq 1: X_{n}=0\right\} \\ q_{i} &:=P_{i}(T=\infty) \end{aligned} $$
(a) By writing $q_{1}$ in terms of $q_{2}$, and $q_{i}$ in terms of $q_{i-1}$ and $q_{i+1}$ for $i \geq 2$, show that $q_{k}=k q_{1}$ for all $k \geq 1$
(b) Hence prove that $X$ is recurrent.
My attempt
a) I have largely proved this using induction but I don't understand why $q_{0}=P_{0}(T=\infty)=P_{0}\left(x_{\infty}=0\right)=0$ . I know that $q_{0} = 0$ because then by the question we have $q_{1}=\frac{1}{2} q_{0}+\frac{1}{2} q_{2}$ which implies that $q_{0} = 0$ and $q_{2} = 2* q_{1}$ which implies $q_{k}=k q_{1}$ as per the question.
b) To prove that X is recurrent, I used the relationship that I proved in a) where $q_{k}=k q_{1}$. Since there are infinite states then $\lim_{k \to \infty} q_{k}=k q_{1} > 1$. This cannot occur as $q_{k}$ is a probability therefore X has to to recurrent.
Am I right? Thank you so much for your help!