Problem in defining an event in Gambler's ruin

In Gambler's ruin problem, suppose you are playing a game. The total money in the game is let's say $S$.

$X_t :=$ Amount of money you have at time $t$ and $X_t \in \{0,1,2, ..., S\}$.

At any time if you win, $X_t = X_{t-1} + 1$ and if you loose, $X_t = X_{t-1} - 1$. The game ends if you either hit $0$ or $S$.

We consider an event $R = \bigcup_{n \geq0} \{X_n = 0\}$ and define Ruin probability as $$Pr = P(R|X_0 = k)$$ for $k \in [0,S]$. I have question regarding this event $R$. Instead of taking a union, shouldn't we take infimum? Because once you reach $0$, you stop. There would not be further any $m(>n)$ s.t. $X_m = 0$.

I believe the convention is that once you hit state $0$ or $S$, you stay in that state forever; i.e. if $X_t = S$ then $X_{t+k} = S$ for all $k \in \{1,2,3,\dots\}$.
• Ok. but if I replace this union with $infimum$ then will their be any issue? – Dark_Knight Feb 1 '18 at 7:39
• I would define the ruin probability as $P( \text{hit } 0 \text{ or hit } S \mid X_0 = k)$ – user365239 Feb 1 '18 at 8:07