Let's take $X_1,X_2,...\sim i.i.d.$ with finite expected value. We have $S_n = \Sigma_{k=0}^{n}$ and filtration $\mathcal{F}_n$ that is generated by process $X_n$,( can we write that $ \forall_n ~ \sigma (X_n)\subset \mathcal{F}_n ) $.
We aim to show that $S_n-nEX_1$ is a martingale with respect to filtration $\mathcal{F}_n$.
It is obvious that $$ E|S_n-nEX_1| \leq E|S_n| + nE|X_1|\leq \infty $$ because expected value is finite.
And also that $S_n$ is measurable wrt $\mathcal{F}_n$ because filtration is generated by $X_n$ and $nEX_1$ is measurable wrt $\mathcal{F}_n$ because it is constant.
There is some troubles with third condition.
$$E( S_n-nEX_1 | \mathcal{F}_{n-1} ) = E(S_n | \mathcal{F}_{n-1}) - E(nEX_1) $$ Firstly we use independence and we get $$ = S_{n-1}+EX_n - nE(EX_1) $$ I have no idea what to do now to deal with the problem.
You are almost done. $S_{n-1}+EX_n - nE(EX_1)=S_{n-1}+EX_1 - nE(EX_1)=S_{n-1}-(n-1)EX_1$.