= Solution
We use <dyadic quadratic variation of Brownian motion>. For fixed rational $0\le u<v$, set $\ell=v-u$ and let $Q_n$ be the sum of the squares of the $2^n$ <Brownian increments> on its equal-length partition. <Independence> and the <normal> <fourth moment> give
$$
\mathbb EQ_n=\ell,\qquad \operatorname{Var}(Q_n)=2\ell^2\,2^{-n}.
$$
For every $\eta>0$, <Chebyshev's inequality> bounds $\mathbb P(|Q_n-\ell|>\eta)$ by $2\ell^2\eta^{-2}2^{-n}$, a summable sequence. The <Borel-Cantelli lemma> implies $Q_n\to\ell$ almost surely. Intersect over positive rational $\eta$ and all rational intervals to obtain one probability-one event on which every such interval has this positive quadratic-variation limit.
If a path were Hölder of exponent $\alpha>1/2$ on a nonempty open interval, choose a compact rational subinterval $[u,v]$ within it and a finite Hölder constant $C$ there. Every partition increment would have absolute value at most $C(\ell2^{-n})^\alpha$, so
$$
Q_n\le C^2\ell^{2\alpha}2^{n(1-2\alpha)}\longrightarrow0.
$$
This contradicts $Q_n\to\ell>0$. The same pathwise event excludes every exponent greater than $1/2$, without needing an uncountable intersection of events. Therefore
$$
\boxed{\text{Almost surely, no nonempty interval supports a Hölder exponent }>\tfrac12.}
$$
This is the <quadratic variation obstruction to Hölder regularity>. It rules out a continuously differentiable trajectory on any interval, since a continuous derivative is bounded on smaller compact intervals and gives a <Lipschitz> bound. By itself it does not rule out differentiability at an isolated point. The stronger <nowhere differentiability of Brownian motion> can also be seen directly as follows.
On a mesh of width $h=2^{-n}$ in $[0,M]$, the probability that any specified three consecutive increments all have absolute value at most $Kh$ is at most $C K^3h^{3/2}$: each increment is $N(0,h)$ and the three are <independent>. There are at most $M/h$ possible blocks, so the probability of any such block is at most $C M K^3h^{1/2}$, summable in $n$. By <Borel-Cantelli lemma>, for every pair of positive integers $M,K$ there are eventually no such blocks. But a finite derivative at some time $t<M$ would imply $|B_s-B_t|\le C_t|s-t|$ near $t$. The three mesh increments beginning at the mesh point immediately to the left of $t$ would then all be at most $6C_th$ for all sufficiently fine meshes. Choosing an integer $K\ge6C_t$ contradicts the preceding event. Thus finite derivatives do not exist anywhere almost surely, including a finite right derivative at $0$. The Hölder estimates below $1/2$ express continuity with a roughness scale; they do not imply differentiability.
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