 Solution

Until the first visit to zero, the positive-side chain is a nearest-neighbour random walk with upward probability $2/3$ and downward probability $1/3$. For an integer $m>1$, let $h_i^{(m)}={\mathrm{Pr}}_i(T_0<T_m)$, $0\le i\le m$. The Markov property givesThe characteristic roots of this recurrence are $1$ and $1/2$. Solving its two boundary equations gives the biased gambler's ruin probabilityA finite path that reaches zero has a finite maximum before doing so, and is counted by $\{T_0<T_m\}$ for every sufficiently large $m$. Conversely each such event implies a visit to zero. Therefore these events increase to $\{T_0<\infty \}$, and continuity of probability givesMore generally the hitting probability from $i>0$ is $2^{-i}$. The finite-boundary derivation avoids choosing an unjustified solution of the infinite recurrence.