= Solution
For every positive real $r$, a centered <normal> variable of <variance> $t-s$ has absolute $r$th moment $c_r|t-s|^{r/2}$, where $c_r=\mathbb E|N(0,1)|^r<\infty$. Fix $0<\alpha<1/2$ and choose $r$ large enough that $\alpha<1/2-1/r$. Then
$$
\mathbb E|B_t-B_s|^r=c_r|t-s|^{1+(r/2-1)}.
$$
The <Kolmogorov continuity theorem> gives a modification with <Hölder continuity> of every exponent less than $(r/2-1)/r=1/2-1/r$, in particular exponent $\alpha$, on each compact interval. This modification and the given continuous <Brownian motion> agree almost surely at all rational times; continuity makes them agree everywhere simultaneously. Thus the property holds for the given trajectories, not just for another version.
Take a <countable> increasing sequence of positive exponents approaching $1/2$, and intersect their probability-one events for every integer time horizon. <Hölder continuity> of a larger exponent implies that of every smaller positive exponent on a compact interval, after adjusting the constant. We have therefore proved the simultaneous assertion
$$
\boxed{\text{Almost surely, }B\text{ is locally }\alpha\text{-Hölder for every }0<\alpha<\tfrac12.}
$$
The compact-interval qualification matters: a single uniform Hölder constant on the entire half-line is not being asserted.
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