= Solution
Fix an integer $k\geq1$ and consider the <independent increments>
$$
\Delta_{j,k}=B_{(j+1)/k}-B_{j/k},\qquad j=0,1,2,\ldots.
$$
For fixed $k$ these are <independent and identically distributed random variables> with the <normal distribution> $N(0,1/k)$, so $p_k=\mathbb P(|\Delta_{j,k}|>1)>0$. The second <Borel-Cantelli lemma> gives $|\Delta_{j,k}|>1$ for infinitely many $j$, <almost surely>. Intersecting these probability-one <events> over the countably many $k$ preserves probability one.
On any path in this intersection, <uniform continuity> on $[0,\infty)$ would give some $\delta>0$ such that $|B_t-B_s|<1$ whenever $|t-s|<\delta$. Choosing $k$ with $1/k<\delta$ contradicts the existence of an increment $|\Delta_{j,k}|>1$. Therefore <Brownian paths are not uniformly continuous on the half-line>:
$$
\boxed{\mathbb P\bigl(B\text{ is uniformly continuous on }[0,\infty)\bigr)=0.}
$$
The event is measurable: for <continuous paths>, <uniform continuity> can be tested by countably many pairs of nonnegative <rational numbers> and tolerances $1/m$. \b[<Brownian motion> has <uniform continuity> on every compact time interval, but <almost surely> fails it on the whole half-line.]
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