Consider a random walk on $\mathbb{Z}$ with increments $X_i$ that are some bounded random variable. Suppose $\mathbb{P}(X>0)$ and $\mathbb{P}(X<0)$ are non-zero, and $\mathbb{E} X\neq 0$. Show that there is some $0<q\neq 1$ so that $q^{S_n}$ is a martingale.
I have reached the point that $\mathbb{E}(q^{S_{n+1}}|\mathbb{F}_n)=\mathbb{E}(q^{S_n}q^{X_{n+1}}|\mathbb{F}_n)=q^{S_n}\mathbb{E}(q^{X})$, thus $q^{S_n}$ is a martingale if and only if $\mathbb{E}(q^{X})=1$.
I tried to use moment generating function $f(t)=\mathbb{E}(e^{tX})$ and show there is some t such that $f(t)=1$. We have $f(0)=1$, Since $X$ is bounded, $\mathbb{E}(X^2)<\infty$, as t goes to $\pm \infty$, $f(t)\to \pm \infty$. Also, $f'(0)=\mathbb{E}(X)\neq 0$, which means around $0$, $f(t)$ is either increasing or decreasing.
But I'm stuck here, how can I show that $f(t)$ can be equal to 1 again for some t?
