Let $S_n$ be a simple random walk: $S_0 = 0, S_n = U_1 + \ldots + U_n$, where $U_i$ are iid such that $0 < P(U_1 = 1) = p < 1$, $P(U_1 = −1) = 1 − p = q$.
Suppose $F_0 = \{\emptyset, \Omega\}$ and $F_n = \sigma(U_1, \ldots, U_n)$.
Let $Z_n = (q/p)^{S_n}$.
(a) Show that $Z_n$ is a positive martingale.
(b) Let $k \in \mathbb{N} ∪ \{0\}$. Show that $$P(\sup_{n\geq0} S_n \geq k) \leq (p/q)^k$$
and that, if $q > p$, $$\mathbb{E}(\sup_{n\geq0} S_n) \leq p/q − p.$$
I managed to solve part a. Please help me with part b