Solution
= Solution
Apply part (b) to $u_n$. Its zero <Dirichlet boundary condition>[boundary value] gives
$$
u_n(x)=-\frac12\mathbb E_x\int_0^T\Delta u_n(B_s)\,ds.
$$
The assumptions say that $\Delta u_n(y)\to0$ for every $y\in D$ and that $|\Delta u_n|\leq C$ uniformly. Thus the integrand converges pointwise to zero and is dominated by $CT$, whose expectation is finite by part (a). The <dominated convergence theorem> gives $u_n(x)\to0$ for every $x\in D$.