= Solution
Use the natural <filtration> $\mathcal F_k=\sigma(X_1,\ldots,X_k)$ and define $M_k=S_k-mk$. The increments $X_k-m$ are centered, independent, and square integrable, so their <conditional expectations> given the past are zero. Thus $(M_k)$ is a <square-integrable> <martingale> with $M_0=0$. <Independence> gives
$$
\mathbb E|M_n|^2=\operatorname{Var}\!\left(\sum_{j=1}^n(X_j-m)\right)=n\sigma^2.
$$
Applying the <Doob L2 maximal inequality> yields
$$
\boxed{\mathbb E\!\left[\max_{0\leq k\leq n}|S_k-mk|^2\right]\leq4n\sigma^2.}
$$
This includes $\sigma^2=0$, when every centered increment vanishes <almost surely>.
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