= Solution
\b[False.] In fact, standard one-dimensional <Brownian motion> satisfies $\liminf_{t\to\infty}B_t=-\infty$ almost surely. For each integer $m\geq1$, let $\tau_{-m}=\inf\{t:B_t=-m\}$. The <Brownian reflection principle>, applied to $-B$, gives
$$
\mathbb P(\tau_{-m}\leq t)=2\mathbb P(B_t\leq-m)=2\bigl(1-\Phi(m/\sqrt t)\bigr)\longrightarrow1,
$$
where $\Phi$ is the <standard normal distribution function>. Thus each negative integer is reached almost surely. Intersecting these countably many probability-one events, all negative integer levels are reached on the same path. Their hitting times must become arbitrarily large, because a continuous path is bounded on every compact time interval. Hence the path cannot tend to positive infinity. Applying the same argument to positive levels also gives $\limsup_{t\to\infty}B_t=+\infty$ almost surely.
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