Random walk's stopped process

Question

If $S_n$ is a simple random walk in $\mathbb{Z}$ starting in $x \in \{1,...,N-1\}$. It is true that $P\{S_{n \wedge T}=y\}=0,$ when $y \in \{0,N\}$? Where $T = \inf\{n: S_n \in \{0,N\}\}$. I can't understand this, could you give me any suggestions to understand this?

Answer 1

Not true. Consider the event $\{S_1 \notin \{0,N\},S_2 \notin \{0,N\},\cdots, S_{n-1} \notin \{0,N\}, S_n=0\}$. This event has positive probability. Under this event we have $T=n$ and $S_{n\wedge T}=S_n=0$ so we get positive probability when $y=0$.