= Solution
Use the <Kolmogorov continuity theorem> in its one-parameter form: if on a compact interval a process satisfies $\mathbb E|X_t-X_s|^p\leq C|t-s|^{1+\beta}$ for some $p,\beta>0$, it has a modification whose paths are <Hölder continuous> of every order $\gamma<\beta/p$ on that interval.
For <Brownian motion>, normal increments give, for every $p>0$,
$$
\mathbb E|B_t-B_s|^p=\mathbb E|Z|^p\,|t-s|^{p/2},\qquad Z\sim N(0,1).
$$
Every such normal moment is finite. Taking $p>2$ yields $\beta=p/2-1$, hence any order below $1/2-1/p$. For a prescribed $0<\alpha<1/2$, choose $p$ with $1/2-1/p>\alpha$.
The continuous modification and the given continuous <Brownian motion> agree at all rational times on one <almost sure event>; continuity makes them agree everywhere on the interval. To obtain all exponents and all compact intervals simultaneously, apply the theorem to integer intervals $[0,N]$ and a countable sequence of positive exponents increasing to $1/2$, then intersect these <almost sure events>. A bound at exponent $\gamma>\alpha$ implies a bound at $\alpha$ on a compact interval. Consequently the <Brownian Hölder regularity> conclusion is
$$
\boxed{|B_t-B_s|\leq C_{N,\alpha}|t-s|^\alpha\quad(s,t\in[0,N]),\qquad0<\alpha<\tfrac12,}
$$
with finite random constants on one common event of probability one. This uses the usual positive-exponent meaning of Hölder continuity.
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