Problem: Let $\{X_i\}$ be i.i.d. random variables with $P(X_i=-1)=P(X_i=1)=\frac{1}{2}$. For $n\geq0$ let $S_n=1+X_1+\cdots+X_n$. Then $S_0,S_1,S_2,\dots$ is a symmetric simple random walk with initial point $S_0=1$. Find the probability that the random walk eventually hits the point $0$. This means that you have to find the probability of the event: $$A=\{\text{there is an }n>0\text{ with }S_n=0\}.$$ [Hint: For each $M>1$ define the event $A_M=\{S_n\text{ hits 0 before hitting }M\}$. Show that $A_M\nearrow A$ and use the continuity of the probability measure.]
My Work: Following the hint, we begin by proving that $A_M\nearrow A$. To prove that $A=\bigcup_{M=2}^\infty A_M$, observe that if $A_M$ holds for a fixed $M>2$, then $S_n$ hits zero before
hitting $M$, and thus it hits zero, so that $A_M\subseteq A$ holds for all $M$, whence $\bigcup_{M=2}^\infty A_M\subseteq A$. Next, if $A$ holds then for some $n>0$ we have $S_n=0$, so
that in particular for all $M>n$ we have that $A_M$ holds, since if $S_n=0$ then $S_n$ hits zero before hitting $M$. Thus, we have $A\subseteq\bigcup_{M=2}^\infty A_M$, so it follows that
$A=\bigcup_{M=2}^\infty A_M$. To see that the events $A_1,A_2\dots$ form an increasing sequence, note that if $A_M$ holds
then $S_n$ hits zero before hitting $M$ so that it also hits zero before hitting $M+1$, so that $A_M\subseteq A_{M+1}$ for all $M>1$.
Now observe that $P(A_M)=\frac{M-1}{M}$ since this is the gambler's ruin problem where $P(A_M)$ gives the probability of loosing if we begin with $1$ dollar and leave after hitting zero or $M$ dollars.
Using the continuity of the probability measure yields
$$P(A)=\lim\limits_{M\to\infty}P(A_M)=\frac{M-1}{M}=1.$$
Do you agree with my work above? Any comments are most welcomed and appreciated.