= Solution
For $m>0$, the <Brownian reflection principle> gives
$$
\mathbb P\left(\sup_{0\leq s\leq t}B_s\geq m\right)
=2\mathbb P(B_t\geq m)\longrightarrow1
$$
as $t\to\infty$. Hence Brownian motion hits every positive integer almost surely. Applying the same argument to $-B$ shows that it hits every negative integer almost surely. Taking the countable intersection of these probability-one events proves
$$
\limsup_{t\to\infty}B_t=+\infty,
\qquad
\liminf_{t\to\infty}B_t=-\infty.
$$
Arbitrarily late positive and negative values occur, and path continuity forces a zero between successive values of opposite sign. Thus <recurrence of one-dimensional Brownian motion> gives infinitely many visits to zero.
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