= Solution
Take a sequence $R\uparrow\infty$ with $R>|x|$. The <events> $\{\tau_r<\tau_R\}$ increase, and their union is $\{\tau_r<\infty\}$. Indeed, if the <sphere> of radius $r$ is reached at a finite time, continuity makes the path bounded up to that time, so a sufficiently large outer <sphere> has not yet been hit. Conversely, each <event> in the union includes a finite inner hit.
Taking the limit of part (a)'s <Brownian sphere-hitting probability in dimension three> yields the sharper formula
$$
\boxed{\mathbb P_x(\tau_r<\infty)=\frac r{|x|}<1\qquad(|x|>r).}
$$
This establishes the required escape probability without presupposing the <transience of Brownian motion in dimension at least three>.
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