Finite-section approximation of a compact operator

If finite-rank orthogonal projections $P_n$ converge strongly to the identity and $A$ is compact, then $\Vert A-P_{n}A{P}_n\Vert \to 0$. The finite-section spectra converge to $\mathrm{Sp}(A)$ in Hausdorff distance, including for nonnormal $A$.