Optional stopping fails for an unbounded simple-random-walk hitting time

For $a\ge 1$, let $E_a$ be the event that the walk hits $1$ before $-a$. Translating the symmetric ruin interval to $\{0,a+1\}$ with starting point $a$, F1 gives $P(E_a)=\frac{a}{a+1}\mathrm{.}$ The $E_a$ increase, and their union is $\{\tau <\mathrm{\infty }\}$: a path that reaches $1$ has a finite minimum before that time and therefore belongs to some $E_a$. Continuity from below gives $P(\tau <\mathrm{\infty })=1$.

By definition, $S_{\tau }=1$ on this probability-one event, whereas $S_0=0$. Thus their expectations differ. This explicitly shows that almost-sure finiteness alone cannot justify passing from $\tau \wedge n$ to $\tau$ in bounded optional sampling, as F2 warns. AC has exactly the inherited role in F3.