 First-passage law of large numbers for an upward skip-free random walk

Suppose an integer-valued random walk starts at $S_0=0$, has upward increments at most one, and satisfies $S_n/n\to \mu >0$ almost surely. Write $T_j$ for its first-passage time to $j\ge 1$. On the event of almost sure convergence, the walk tends to $+\infty$ and cannot jump over any positive integer, so every $T_j$ is finite. The bound $S_n\le n$ gives $T_j\ge j\to \infty$. Along this subsequence, $j/T_j=S_{T_j}/T_j\to \mu$ on each such sample path, proving the displayed reciprocal limit. No independence between the hitting times and the path is required. The strong law of large numbers supplies the hypothesis for integrable independent and identically distributed random variables as steps with positive mean.