= Solution
The increment variance is
$$
\operatorname{Var}(\xi_k)=1-(2p-1)^2=4p(1-p),
$$
so independence gives
$$
\operatorname{Var}(S_n)=4np(1-p).
$$
Linear interpolation makes the supremum of the absolute centered process occur at an integer time. The <Doob L2 maximal inequality> therefore gives
$$
\begin{aligned}
\mathbb E\sup_{0\leq t\leq1}|S_t^{(n)}-\mu t|^2
&=\frac1{n^2}\mathbb E\max_{0\leq k\leq n}|M_k|^2\\
&\leq\frac4{n^2}\mathbb E M_n^2
=\frac{16p(1-p)}n
\leq\boxed{\frac4n}.
\end{aligned}
$$
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