Infinite mean first passage of a simple symmetric random walk ($\mathbb{E}{T}_1=\infty ,\mathbb{P}(T_1<\infty )=1$)

For a simple symmetric random walk starting from zero, its first hitting time of $1$ is almost surely finite but has infinite expectation. Stopping at exit from $[-m,1]$, the optional stopping theorem for $S_n$ gives upper-exit probability $m/(m+1)$, while stopping $S_n^2-n$ gives mean exit time $m$. The first probabilities tend to one, and the exit times are bounded above by the first passage to $1$, proving both conclusions. Finite-time stopping and bounded stopped positions justify passage to the exit time.