Consider a random walk $S_n=S_0+\sum^b_{i=1}X_i$ with i.i.d steps $X_i$ taking value $4$ and $-7$ with probabilities $\frac{7}{11}$ and $\frac{4}{11}$ respectively.
I would like to show that $S_n$ is a martingale.
My attempt:
$E[X]=(4)\frac{7}{11}+(-7)\frac{4}{11}$
$E[S_{n+1}|\mathcal{F}_n]=E[S_n+X_{n+1}|\mathcal{F}_n]=S_n+E[X]=S_n$
Where it was used that $X_i$ are iid random variables.
Is this correct?