I never wrapped my head around when "the probability converges to 1" is enough as opposed to being strictly "the probability is 1".
In particular, to use Doob's theorem, given a martingale $M_n$ and a stopping time $T$, $\mathbb{E}(M_T)=\mathbb{E}(M_0)$, if $\mathbb{P}(T<\infty)=1$ and $|M_n|\leq k\in \mathbb{R}, \forall n$.
But what if I can show that $\mathbb{P}(T>\infty)$ converges to 0? Does that imply $\mathbb{P}(T<\infty)=1$?
In the SSRW example, we get that $\mathbb{P}_1(V_0>V_N)=\frac{1}{N}$, but this doesn't imply that $\mathbb{P}_1(V_0<\infty)=1$, is this due to convergence is not enough to use Doob's theorem, or because "we might get lost forever in states between 0 and $N$"?