Suppose $X_n=q$ with probability $p$, and $X_n=-p$ with probability $q$ where $p+q=1$. Prove that for every $n$, the probability that $S_k\geq b$ for any $k$ as $1\leq k \leq n$ is at most $e^{-b^2/2n}$. Of course $S_n=\sum_{i=1}^k X_i$. The problem was formulated with gains of a gambler where at each round he can make $X_n$, and we need to bound the probability that the total gain in any of the first $n$ rounds exceeds n with the specified bound.
I tried using the martingale Sn and stopping time $T=\min\{\min_k\{S_k>=b\},n\}$ but couldn't get that bound. I know i need to use Hoeffding's inequality, but it relates only to the last sum, $S_n$.