= Solution
Choose $0<\alpha<\varepsilon/p$. At dyadic level $m$, <Markov inequality> and a union bound give
$$
\mathbb P\left(\max_k|X^n_{(k+1)2^{-m}}-X^n_{k2^{-m}}|>C2^{-m\alpha}\right)
\leq cC^{-p}2^{-m(\varepsilon-\alpha p)},
$$
uniformly in $n$. Summing over $m$ shows that, outside a set of probability at most $C_0C^{-p}$, all dyadic increments obey this bound. Chaining dyadic approximations and using continuity gives
$$
|X_t^n-X_s^n|\leq C_1C|t-s|^\alpha
$$
for all $s,t$. Since $X_0^n=0$, the paths then lie in the stated compact Hölder set by the <Arzela-Ascoli theorem>. Taking $C$ large proves tightness.
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