= Solution
Sample <Brownian motion> at integer times and put $Z_n=B_n/\sqrt n$. Every $Z_n$ has <normal distribution> $N(0,1)$, although the $Z_n$ are correlated. Let
$$
L=\limsup_{n\to\infty}Z_n.
$$
For every fixed integer $m$,
$$
L=\limsup_{n\to\infty}\frac{B_n-B_m}{\sqrt n},
$$
because $B_m/\sqrt n\to0$ <almost surely>. The right-hand expression depends only on the future <independent increments> $B_{j}-B_{j-1}$ with $j>m$. Thus $\{L\geq a\}$ is, up to a <null set>, a <tail event> of those <independent increments>, and its probability is zero or one by the <Kolmogorov zero-one law>.
For any finite real $a$, every $Z_n$ exceeds $a$ with the same positive probability $p_a=\mathbb P(N(0,1)\geq a)$. For each $N$,
$$
\mathbb P\left(\bigcup_{n\geq N}\{Z_n\geq a\}\right)\geq p_a.
$$
Taking the decreasing intersection over $N$ shows that $Z_n\geq a$ infinitely often with probability at least $p_a$. This event implies $L\geq a$, so $\mathbb P(L\geq a)>0$. The zero-one law makes it one. Intersecting over positive integers $a$ gives $L=+\infty$ <almost surely>. The continuous-time limit superior is at least the one along integers, hence
$$
\boxed{\limsup_{t\to\infty}\frac{B_t}{\sqrt t}=+\infty
\quad\text{almost surely}.}
$$
This proves that <Brownian fluctuations exceed the square-root scale> without needing the lower bound in the law of the iterated logarithm.
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