= Solution
For $a>0$, the <Brownian reflection principle> at the <first-passage time> of $a$ gives the <Brownian running maximum> identity
$$
\mathbb P(S_t\geq a)=2\mathbb P(B_t\geq a)=2\left(1-\Phi\left(\frac a{\sqrt t}\right)\right),\qquad t>0,
$$
where $\Phi$ is the <distribution function> of the standard <normal distribution>. Indeed, reflection pairs paths that have reached $a$ and end below $a$ with paths ending above $a$; $B_t$ has no atom at $a$.
As $t\to\infty$, this probability tends to one. The <events> $\{S_n\geq a\}$ increase with integer $n$, so with probability one the path reaches $a$ in finite time. Taking a countable intersection over positive integer $a$ gives
$$
\boxed{\mathbb P(S_\infty=\infty)=1.}
$$
Here $S_\infty=\sup_{s\geq0}B_s$. This also proves that every positive-level <Brownian first-passage time> is finite <almost surely>.
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