= 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> gives
$$
R(\zeta,x_n)-R(\zeta,x)=R(\zeta,x_n)(x_n-x)R(\zeta,x).
$$
The resolvents therefore converge uniformly on $\Gamma$. Both <holomorphic functional calculus> values use that same contour, so
$$
\|\Theta_{x_n}(f)-\Theta_x(f)\|\leq\frac{\operatorname{length}(\Gamma)}{2\pi}\sup_\Gamma|f|\sup_\Gamma\|R(\zeta,x_n)-R(\zeta,x)\|\longrightarrow0.
$$
Hence \b[$\Theta_{x_n}(f)\to\Theta_x(f)$] for each fixed $f\in\mathcal O(U)$.
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