Approximation of approximation numbers by truncation

Let A be a bounded linear operator on some infinite-dimensional separable Hilbert space H and let A(n) be the orthogonal compression of A to the span of the first n elements of an orthonormal basis of H. Ne show that, for each k greater than or equal to 1, the approximation numbers s(k)(A(n)) converge to the corresponding approximation number s(k)(A) as n --> infinity. This observation implies almost at once some well known results on the spectral approximation of bounded selfadjoint operators. For example, it allows us to identify the limits of all upper and lower eigenvalues of A(n), in the case where A is selfadjoint. These limits give us all points of the spectrum of a selfadjoint operator which lie outside the convex hull of the essential spectrum. Moreover, it follows that the spectrum of a selfadjoint operator A with it connected essential spectrum can be completely recovered from the eigenvalues of A(n) as n goes to infinity.