Let us consider the Boundary Value Problem (BVP) for the discrete Dirac equations , , , , where , , and , are complex sequences, , , and λ is an eigenparameter. Discussing the eigenvalues and the spectral singularities, we prove that this BVP has a finite number of eigenvalues and spectral singularities with a finite number of multiplicities, if , holds, for some and . MSC:34L40, 47A10, 47A75.