= Solution
\b[True.] The unit-time increments $Z_k=B_k-B_{k-1}$ are <independent and identically distributed random variables> with standard <normal distribution>, hence integrable with mean zero. The <strong law of large numbers> gives $B_n/n\to0$ almost surely at integer times.
To control the intervals between integers, put $M_n=\sup_{0\leq u\leq1}|B_{n+u}-B_n|$. <Stationary increments> and the <Doob L2 maximal inequality> give $\mathbb E M_n^2\leq4$. Thus the <Markov inequality> yields
$$
\sum_{n=1}^\infty\mathbb P(M_n>\varepsilon n)\leq\frac4{\varepsilon^2}\sum_{n=1}^\infty n^{-2}<\infty.
$$
The <Borel-Cantelli lemma>, followed by a countable intersection over positive rational $\varepsilon$, implies $M_n/n\to0$ almost surely. For $n\leq t\leq n+1$,
$$
\frac{|B_t|}{t}\leq\frac{|B_n|+M_n}{n}.
$$
Both terms tend to zero on the same probability-one event, proving $\boxed{B_t/t\to0\text{ almost surely}}$ for continuous time as well.
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