Summary: We consider sequences of Hermitian matrices with increasing dimension, and we provide a general tool for determining the asymptotic spectral distribution of a ‘difficult’ sequence \(\{A_n\}_n\) from the one of ‘simpler’ sequences \(\{B_{n,m}\}_n\) that approximate \(\{A_n\}_n\) when \(m \rightarrow \infty\). The tool is based on the notion of an approximating class of sequences (a.c.s.), which was inspired by the work of P. Tilli [Linear Algebra Appl. 278, No.  1–3, 91–120 (1998; Zbl 0934.15009)] and the second author [Linear Algebra Appl. 328, No. 1–3, 121–130 (2001; Zbl 1003.15008)], and it is applied here in a more general setting. An a.c.s.-based proof of the famous Szegő theorem on the spectral distribution of Toeplitz matrices is finally presented.