  This paper is concerned with regular approximations of spectra of singular
discrete linear Hamiltonian systems with one singular endpoint. For any given
self-adjoint subspace extension (SSE) of the corresponding minimal subspace,
its spectrum can be approximated by eigenvalues of a sequence of induced
regular SSEs, generated by the same difference expression on smaller finite
intervals. It is shown that every SSE of the minimal subspace has a pure
discrete spectrum, and the k-th eigenvalue of any given SSE is exactly the
limit of the k-th eigenvalues of the induced regular SSEs; that is, spectral
exactness holds, in the limit circle case. Furthermore, error estimates for the
approximations of eigenvalues are given in this case. In addition, in the limit
point and intermediate cases, spectral inclusive holds.
