Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 2 / 27J / d / Solution

For $a>0$, the Brownian reflection principle gives $\mathbb{P}({\sup }_{0\le s\le T}{B}_s\ge a)=2\mathbb{P}(B_T\ge a)$. 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. ThereforeBy 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 $\boxed{\mathbb{P}(Z=+\infty )=1}$.