Let \(A\) be a linear bounded operator acting on a Hilbert space, and let \(P_n\) be an increasing sequence of orthogonal projections of rank equal to \(n\), respectively. The present note is concerned with the convergence of various spectral invariants of \(A_n=P_nAP_n\) to those of \(A\). By remarking that the approximation numbers \(s_k(A_n)\) converge to \(s_k (A)\), the authors derive very powerful consequences, for instance for a self-adjoint limit \(A\). The same ideas have appeared earlier in the theory of Padé approximation, see for instance \textit{G. A. Baker, Jr} and \textit{P. Graves-Morris} [``Padé Approximants'' (Addison-Wesley, Reading Mass.) (1981; Zbl 0468.30032 and Zbl 0468.30033)].