Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 201 / 5 / c / Solution

Apply part (b) to $u_n$. Its zero boundary value givesThe assumptions say that $\Delta u_n(y)\to 0$ for every $y\in D$ and that $|\Delta u_n|\le C$ 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 $u_n(x)\to 0$ for every $x\in D$.