= Solution
Let $(B_t)$ be <Brownian motion> in $\mathbb R^3$ started at a nonzero point. The function $h(x)=|x|^{-1}$ is a positive <harmonic function> on $\mathbb R^3\setminus\{0\}$ because its <Laplace operator>[Laplacian] vanishes there. Stopping on annuli and applying <Itô formula> shows that $h(B_t)$ is a <local martingale>. Letting the inner boundary shrink to zero also shows that three-dimensional Brownian motion does not hit the origin.
A positive local martingale is a <supermartingale>, so $h(B_t)$ is $L^1$-bounded by $h(B_0)$. The upcrossing proof from part (c), which applies verbatim to a positive supermartingale, therefore gives a finite almost-sure limit
$$
|B_t|^{-1}\longrightarrow Y.
$$
Solved by gpt-5.6-sol high.
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