= Solution
First construct a centred <Gaussian process> on the countable dyadic set with <covariance function> $\mathbb E(B_sB_t)=\min(s,t)$. These consistent finite-dimensional <normal distributions> exist because, for coefficients $a_j$,
$$
\sum_{i,j}a_ia_j\min(t_i,t_j)=\int_0^1\left(\sum_ja_j\mathbf1_{\{u\leq t_j\}}\right)^2du\geq0;
$$
the <Kolmogorov extension theorem> realizes them on one <probability space>. Every increment has the <normal distribution> $N(0,|t-s|)$, so for every finite $p\geq1$,
$$
\|B_t-B_s\|_p=(\mathbb E|Z|^p)^{1/p}|t-s|^{1/2}.
$$
Use the preceding estimate with $\beta=1/2$, $p>2$ and $0<\alpha<1/2-1/p$. It yields an <almost surely> <Hölder continuous> extension from the dense dyadic set to $[0,1]$. Approximate arbitrary times by dyadic times; continuity of the paths and of the Gaussian <covariance function> preserves all finite-dimensional laws. Disjoint increments are jointly Gaussian with zero cross-covariances, hence <independent>, with the required variances. Together with $B_0=0$ and continuity, this constructs <Brownian motion>.
For any $\alpha<1/2$, choose $p$ large enough. Taking a countable increasing sequence of exponents tending to $1/2$ gives \b[simultaneous Hölder regularity of every order $\alpha<1/2$] on a common probability-one set; the same argument on each compact time interval gives local <Brownian Hölder regularity>. This method does not give the endpoint exponent. In fact that endpoint fails: if a path had a finite $1/2$-<Hölder seminorm> $K$, all its $2^n$ adjacent normalized increments at level $n$ would have magnitude at most $K$. These are <independent> standard <normal random variables>, so for a fixed integer $m$ the probability that their maximum is at most $m$ is $\mathbb P(|Z|\leq m)^{2^n}\to0$. The event of a finite $K$ is contained in the union over $m$ of events with this bound at every level, each of probability zero. Thus \b[Brownian paths are not $1/2$-Hölder continuous on $[0,1]$ almost surely].
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