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Continue with $A_d$ equal to the integral defined in the PDF and $u(z)=|z{|}^{2-d}$. Until the Brownian motion hits the radius-$\epsilon$ ball, $u(B_s)$ is a bounded local martingale, hence a true martingale. At the finite stopping time $t\wedge T_{\epsilon }$, optional stopping givesRearranging,The fixed random variable $u(B_t)$ is integrable by the Gaussian heat kernel and the local-integrability calculation above. Also ${\mathbb{P}}_x(T_{\epsilon }\le t)\le (\epsilon /|x|{)}^{d-2}\to 0$. For any $R>0$, the final expectation is at mostFirst send $\epsilon \downarrow 0$, then $R\to \infty$, proving that it vanishes. The identities from part (b) giveSubtracting yields the small-ball Brownian hitting asymptoticThus the constant in this printed limit requires exactly the same reciprocal correction as in part (b). For $d=3$ the multiplier is $2\pi$, not $1/(2\pi )$. The argument proves the corrected finite-time limit directly, rather than inferring it from the eventual hitting probability alone.