give a simple symmetric random walk, denote $\mathbb P_0(T_1 < \infty)$ the probability that starting at $0$ and the first time reaching level 1 within finite discrete time steps. What is $\mathbb P_0(T_1 < \infty)$?
So can I assume that it hits level $1$ within $n$ steps, then use the hitting time thm to calculate the probability? Or is there another way to this?
The probability you're asking is 1.
In fact, the probability of reaching any point within a finite number of steps is always 1.
An intuitive way to think about it is: " Any finite sequence will appear infinitely often"