= Solution
For $a>0$, the <Brownian reflection principle> gives $\mathbb P(\sup_{0\leq s\leq T}B_s\geq a)=2\mathbb P(B_T\geq a)$. Reflection of the path after its first hit of $a$ preserves its law and pairs paths ending below $a$ with paths ending above $a$, proving this identity. Therefore
$$
\mathbb P\left(\sup_{0\leq s\leq T}B_s\leq a\right)=2\Phi(a/\sqrt T)-1\longrightarrow0.
$$
By monotone increase of the finite-time suprema, the <probability> that the full supremum is at most any finite $a>0$ is zero. Taking a countable union over <integer> $a$ gives $\boxed{\mathbb P(Z=+\infty)=1}$.
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