Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 10 / 2 / b / Solution

Fix an admissible contour $\Gamma$ with interior $D$ compactly contained in $U$ and containing $\sigma (x)$. Apply the preceding spectral inclusion to $D$, so that it also contains $\sigma (x_n)$ for large $n$. Uniformly on the compact contour, the Neumann series makes $R(\zeta ,x_n)$ bounded, and the resolvent identity givesThe resolvents therefore converge uniformly on $\Gamma$. Both holomorphic functional calculus values use that same contour, soHence ${\Theta }_{x_n}(f)\to {\Theta }_x(f)$ for each fixed $f\in \mathcal{O}(U)$.